7^^   ^,  £^^ 


IN  MEMORIAM 
FLORIAN  CAJORl 


d 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation  • 


http://www.archive.org/details/fisharithmeticOfishrich 


.ah'tnson'B   ^\iaxitx   Course. 


THE 


COMPLETE 


AEITHMETIC. 


ORAL   AND    WRITTEl^. 


SECOND    PART. 


By  DANIEL  W.   FISH,   A.M., 

▲UTHOB   OP   BOBINSON'S    8BEIES    OP    PBOGRBSSIVE    ABITHlfETICB 


IVISON,  BLAKEMAN,  TAYLOR  &  CO., 
-      NEW  YORK  AND  CHICAGO 

1881. 


EOBINSON'S 

Shorter  Course. 


FIJ^ST  BOOK  IN-  ARITHMETIC,  Primary, 
COMPLETE  ARITHMETIC.  In  One  volume,^ 
COMPLETE  ALGEBRA. 

ARITHMETICAL  PROBLEMS,    Oral  and  Written. 
ALGEBRAIC  PROBLEMS. 

KE  YS  to  Complete  Arithmetic  and  Problems,  and 

to  Complete  Algebra  and  Problems, 

in  separate  volumes,  for  Teachers, 


Arithmetic,  ouaz  and  wkitten,  usually  taught  in 
THREE  hooks,  is  now  offered,  complete  and  thorough, 
in  ONE  booh,  "  the  complete  arithmetic:' 

*  This  Complete  Arithmetic  is  also  published  in  two  volumes.  PAMT  Xm 
and  TA:RT  II,  are  each  bound  separately^  and  in  cloth. 


Copyright,  1874,  by  DANIEL  W.  FISH. 


JDlectrotyped  by  Smith  &  McDougal,  82  Beekmau  St.,  N.  Y. 


PREFACE 


rpHE  design  of  the  author,  in  the  preparation  of  the  Completb 
-^  Arithmetic,  has  been  to  furnish  a  text-book  on  the  subject  of 
arithmetic,  complete  not  only  as  a  treatise,  but  as  a  comprehensive 
manual  for  the  class-room,  and,  therefore,  embodying  every  necessary 
form  of  illustration  and  exercise,  both  oral  and  written.  Usually, 
this  subject  has  been  treated  in  such  a  way  as  to  form  the  contents 
of  three  or  more  graded  text-books,  the  oral  exercises  being  placed  in 
a  separate  volume.  In  the  present  treatise,  however,  the  whole 
subject  is  presented  in  all  its  different  grades  ;  and  the  oral,  or  men- 
tal, arithmetic,  so  called,  has  been  inserted,  where  it  logically  and 
properly  belongs,  either  as  introductory  to  the  enunciation  of  prin- 
ciples or  to  the  statement  of  practical  rules — the  treatment  of  every 
topic  from  the  beginning  to  the  end  of  the  book  being  thoroughly 
inductive. 

In  this  way,  and  by  carefully  constructed  analyses,  applied  to  all 
the  various  processes  of  mental  arithmetic,  the  pupiFs  mind  cannot 
fail  to  become  thoroughly  imbued  with  clear  and  accurate  ideas  in 
respect  to  each  particular  topic  before  he  is  required  to  learn,  or 
apply  to  written  examples,  any  set  rule  whatever.  The  intellect  of 
the  pupil  is  thus  addressed  at  every  step ;  and  every  part  of  the 
instruction  is  made  the  means  of  effecting  that  mental  development 
which  constitutes  the  highest  aim,  as  well  as  the  most  important 
result,  of  every  branch  of  education. 

This  mode  of  treatment  has  not  only  the  advantage  of  logically 
training  the  pupil's  mind,  and  cultivating  his  powers  of  calculation, 
but  must  also  prove  a  source  of  economy,  both  of  time  and  money, 
inasmuch"  as  it  is  the  means  of  substituting  a  single  wlume  for  an 
entire  aeries  of  text-hooks. 


IV  PREFACE. 

As  the  time  of  many  pupils  will  not  permit  them  to  pursue  this 
study  through  all  of  its  departments,  the  work  is  issued  in  two  parts, 
as  well  as  in  a  single  'colume.  This  will,  it  is  thought,  be  also  con- 
venient for  graded  schools^  in  supplying  a  separate  book  for  classes 
of  the  higher  and  lower  grades  respectively^  without  requiring  any 
unnecessary  repetition  or  review. 

In  this,  the  Second  Part,  all  the  higher  departments  of  arith- 
metic including  Mensuration  are  presented,  commencing  with  Per- 
centage,  the  study  of  which  can  be  taken  up  by  the  pupil  imme- 
diately on  completing  the  First  Part.  This  part  of  the  subject 
has  been  treated  in  a  comprehensive  manner,  and  is,  in  all  respects, 
adapted  to  the  wants  of  the  present  time,  recognizing  and  explain- 
ing all  the  recent  changes  in  Custom-house  Business,  Exchange,  etc. 

An  Appendix  of  forty- eight  pages  of  valuable  Tables  and  Problems 
has  been  added  to  this  pa/rt  of  the  work,  containing  much  useful 
and  practical  information,  fresh  and  important^  obtained  by  much 
labor  of  research  and  inquiry,  which,  with  many  other  improve- 
ments, particularly  adapt  this  work  to  the  wants  of  the  student 
qualifying  for  business,  and  of  graduating  classes  in  High  Schools 
and  Academies,  as  well  as  of  Mercantile  and  Commercial  colleges. 

The  Beviews  interspersed  throughout  the  book  will  be  found  to 
be  just  what  is  needed  by  the  student  to  make  his  progress  sure 
at  each  step,  and  to  give  him  comprehensive  ideas  of  the  subject 
as  he  advances.  Carefully  constructed  Synopses  have  also  been 
inserted,  with  the  view  to  afford  to  both  teacher  and  pupil  a  ready 
means  of  drill  and  examination,  as  well  as  to  present,  in  a  clear, 
concise,  and  logical  manner,  the  relations  of  all  the  different  depart- 
ments of  the  subject,  with  their  respective  sub4opics,  definitions, 
principles,  and  rules. 

It  is  confidently  believed  that,  on  examination,  the  work  as  a 
whole,  as  well  as  in  its  separate  parts,  will  commend  itself  to  teach- 
ers and  others,  by  the  careful  grading  of  its  topics  ;  the  clearness 
and  conciseness  of  its  definitions  and  rides ;  its  improved  methods 
of  analysis  and  operation;  the  great  number  and  variety  of  its 
examples,  both  oral  and  written,  embodying  and  elucidating  all  the 
ordinary  business  transactions ;  and  in  the  omission  of  all  obsolete 


PREFACE.  V 

teTfns  and  discarded  usages,  as  well  as  in  the  introduction  of  many 
novel  features  favorable  both  to  clearness  and  brevity. 

Great  oains  have  also  been  taken  to  make  this  work  superior  to 
all  others  in  its  typographical  arrangement  and  finish,  and  in  the 
general  tastef  ulness  of  its  mechanical  execution. 

The  author  takes  pleasure  in  acknowledging  his  indebtedness  for 
many  valuable  suggestions  received  from  teachers  of  experience 
and  others  interested  in  the  work  of  education ;  especially  to  Joseph 
Ficklin,  Ph.  D.,  Professor  of  Mathematics  in  the  University  of 
Missouri,  by  whom  chiefly  the  sections  upon  Involution,  Evolution, 
Progressions,  and  Annuities  have  been  prepared ;  as  well  as  to 
Henry  Kiddle,  A.  M. ,  Superintendent  of  Schools  in  the  city  of  New 
York,  for  valuable  assistance,  especially  in  the  higher  departments 
of  Percentage,  and  for  important  suggestions  in  relation  to  other 
parts  of  the  work. 

D.  W.  F. 

Bbookltn,  January^  187S. 


"TN  order  to  teach  any  subject  with  the  best  success,  the  instruc- 
-^  tor  should  not  only  fully  understand  it,  in  all  its  principles  and 
details,  but  should  also  clearly  perceive  what  particular  faculties 
of  the  mind  are  concerned  in  its  acquisition  and  use. 

Arithmetic  is  pre-eminently  a  subject  of  practical  value  ;  that  is, 
it  is  one  to  be  constantly  applied  to  the  practical  affairs  of  life. 
But  this  is  true  only  in  a  limited  sense.  Very  few  ever  need  to 
apply  to  any  of  the  purposes  of  business  more  than  a  small  part  of 
the  principles  and  rules  of  calculation  taught  in  the  text-books. 
Every  branch  of  business  has  its  own  requirements  in  this  respect, 
and  these  are  all  confined  within  very  narrow  limits. 

The  teaching  of  arithmetic  must,  therefore,  to  a  great  extent,  be 
considered  as  disciplinary, — as  training  and  developing  certain 
faculties  of  the  mind,  and  thus  enabling  it  to  perform  its  functions 
with  accuracy  and  dispatch.  The  following  svggestions,  having 
reference  to  this  twofold  object  of  arithmetical  instruction  are  pre- 
sented to  the  teacher,  as  a  partial  guide,  not  only  in  the  use  of  this 
text-book,  but  in  the  treatment  of  the  subject  as  a  branch  of 
education. 

Seek  to  cultivate  in  the  pupil  the  habit  of  self-reliance.  Avoid 
doing  for  him  anything  which,  either  with  or  without  assistance, 
he  should  be  able  to  do  for  himself.  Encourage  and  stimulate  his 
exertions,  but  do  not  supersede  them. 

Never  permit  him  to  accept  any  statement  as  true  which  he  does 
not  understand.  Let  him  learn  not  by  authority  but  by  demonstra- 
tion addressed  to  his  own  intelligence.  Encourage  him  to  ask 
questions  and  to  interpose  objections.  Thus  he  will  acquire  that 
most  important  of  all  mental  habits,  that  of  thinking  for  himself. 


SUGGESTIOJS^S     TO     TEACHERS,  VU 

Carefully  discriminate,  in  the  instruction  and  exercises,  as  to 
wliich  faculty  is  addressed, — whether  that  of  analysu  or  reasoning, 
or  that  of  calculation.  Each  of  these  requires  peculiar  culture,  and 
each  has  its  appropriate  period  of  development.  In  the  first  stage 
of  arithmetical  instruction,  calculation  should  be  chiefly  addressed, 
and  analysis  or  reasoning  employed  only  after  some  progress  has 
been  made,  and  then  very  slowly  and  progressively.  A  young 
child  will  perform  many  operations  in  calculation  which  are  far 
beyond  its  powers  of  analysis  to  explain  thoroughly. 

In  the  exercise  of  the  calculating  faculty,  the  examples  should  be 
rapidly  performed,  without  pause  for  explanation  or  analysis  ;  and 
they  should  have  very  great  variety,  and  be  carefully  arranged 
so  as  to  advance  from  the  simple  and  rudimental  to  the  complicated 
and  diflBlcult. 

In  the  exercise  of  the  analytic  faculty,  great  care  should  be  taken 
that  the  processes  do  not  degenerate  into  the  mere  repetition  of 
formulcB,  These  forms  of  expression  should  be  as  simple  and  con* 
cise  as  possible,  and  should  be,  as  far  as  practicable,  expressed  in 
the  pupil's  own  language.  Certain  necessary  points  being  attended 
to,  the  precise  form  of  expression  is  of  no  more  consequence  than 
any  particular  letters  or  diagrams  in  the  demonstiation  of  geomet- 
rical theorems.  Of  course,  the  teacher  should  carefully  criticise 
the  logic  or  reasoning,  not  so  as  to  discourage,  but  still  insisting 
upon  perfect  accuracy/  from  the  first. 

The  oral  or  mental  arithmetic  should  go  hand  in  hand  with  the 
written.  The  pupil  should  be  made  to  perceive  that,  except  for  the 
difficulty  in  retaining  long  processes  in  the  mind,  all  arithmetic 
ought  to  be  oral,  and  that  the  slate  is  only  to  be  called  into  requi- 
sition to  aid  the  mind  in  retaining  intermediate  processes  and 
results.  The  arrangement  of  this  text  book  is  particularly  favora- 
ble for  this  purpose. 

BeflnitioTis  and  principles  should  be  carefully  committed  to 
memory.  No  slovenliness  in  this  respect  should  be  permitted.  A 
definition  is  a  basis  for  thought  and  reasoning,  and  every  word 
which  it  contains  is  necessary  to  its  integrity.  A  child  should  not 
be  expected  to  frame  a  good  definition.     Of  course,  the  pupil  should 


Tlli  SUGGESTIOIJ^S     TO     TEACHERS. 

be  required  to  examine  and  criticise  the  definitions  given,  since  this 
will  conduce  to  a  better  understanding  of  their  full  meaning. 

In  conducting  recitations,  the  teacher  should  use  every  means 
that  will  tend  to  awaken  thought.  Hence,  there  should  be  great 
variety  in  the  examples,  both  as  to  their  construction  and  phrase- 
ology, so  as  to  prevent  all  mechanical  ciphering  according  to  fixed 
methods  and  rules. 

The  Rules  and  FormulcB  given  in  this  book  are  to  be  regarded  as 
summaries  to  enable  the  pupil  to  retain  processes  previously  ana- 
lyzed and  demonstrated.  They  need  not  be  committed  to  memory, 
since  the  pupil  will  have  acquired  a  sufficient  knowledge  of  the 
principles  involved  to  be  able,  at  any  time,  to  construct  rules,  if  he 
has  properly  learned  what  precedes  them. 

In  the  higher  department  of  arithmetic,  the  chief  difficulty  con- 
Bists  in  giving  the  pupil  a  clear  idea  of  the  nature  of  the  business 
transactions  involved.  The  teacher  should,  therefore,  strive  by 
careful  elucidation,  to  impart  clear  ideas  of  these  transactions  before 
requiring  any  arithmetical  examples  involving  them  to  be  per- 
formed. When  the  exact  nature  of  the  transaction  is  understood, 
the  pupil's  knowledge  of  abstract  arithmetic  will  often  be  sufficient 
to  enable  hiTn  to  solve  the  problem  without  any  special  rule. 

The  teacher  should  be  careful  not  to  advance  too  rapidly.  The 
mind  needs  time  to  grasp  and  hold  firmly  every  new  case,  and  then 
additional  time  to  bring  its  new  acquisition  into  relation  with  those 
preceding  it.  Hence  the  need  of  frequent  reviews,  in  order  to  give 
the  pupil  a  comprehensive  as  well  as  an  accurate  and  permanent 
knowledge  of  this  subject. 

The  Synopses  for  Beview  interspersed  through  this  work,  are  de- 
signed to  afford  assistance  to  the  teacher  in  accomplishing  this  object. 
Each  of  these  Synopses  exhibits  a  brief,  but  definite,  summary  of  all 
that  is  treated  under  the  particular  topic  referred  to,  systematically 
and  logically  arranged,  showing  not  only  the  different  sub-topics,  and 
their  relations  to  each  other  and  the  general  subject,  but  also  the 
necessary  preliminary  definitions.  Thus  the  teacher  wiU  be  able 
readily  to  ask  an  exhaustive  series  of  questions,  without  having 
recourse  to  every  paragraph  and  page  preceding. 


SUGGESTIOKS     TO     TEACHERS.  IX 

Various  useful  exercises  may  be  based  upon  these  synopses. 
After  the  pupil  has  become  familiar  with  their  mode  of  construc- 
tion, he  may  be  required  to  write  out,  from  memory,  an  outline 
synopsis  of  each  section  that  he  has  studied,  so  as  to  show  whether 
or  not  he  has  comprehended  the  relations  of  the  various  parts  of  the 
subject  which  he  has  passed  over.  Or  the  whole  or  a  part  of  a 
Synopsis,  embracing  one  or  more  topics,  may  be  placed  upon  the 
blackboard,  and  the  pupil  required  to  give  briefly  but  accurately 
the  sub-divisions,  definitions,  principles,  etc.,  involved  in  each.  By 
this  means,  if  further  tested  by  questions,  a  thorough  and  well- 
classified  knowledge  of  the  whole  subject  will  be  permanently 
impressed  upon  his  mind- 
Editions  of  this  book  are  bound  with  and  witJiout  answers.  Those 
with  answers  will  be  sent,  unless  otherwise  ordered. 


artfiV'^abk'iitraB 


PAGB 

Percentage. 1 

Profit  and  Loss 13 

Commission 20 

Synopsis 28 

Interest 29 

Problems  in  Interest. ....  40 

Compound  Interest 45 

Annual  Interest 48 

Partial  Payments 50 

Discount 54 

Bank  Discount 57 

Savings  Bank  Accounts  . .  62 

Synopsis 65 

Stocks 66 

Insurance 76 

Life  Insurance 80 

Taxes 84 

Synopsis 88 

Exchange 89 

Foreign  Exchange 93 

Arbitration  of  Exchange. .  98 

Custom-house  Business. . .  102 

Equation  of  Payments 105 

Averaging  Accounts 110 

Synopsis 118 

Ratio 119 

Proportion 123 

Simple  Proportion 126 

Compound  Proportion. ...  131 


PAG  1 

Partnership 137 

Alligation 143 

Synopsis 149 

Test  Problems. 150 

Involution 155 

EVOLLT^ION, 161 

Cube  Root 168 

Roots  of  Higher  Degree. .  174 
Arithmetical  Pr(^ression.  175 
Geometrical  Progression..  180 

Annuities 185 

Synopsis 189 

Mensuration. 190 

Triangles. 191 

Quadrilaterals 195 

Circles. 197 

Similar  Plane  Figures 199 

Solids 203 

Prisms. 203 

Pyramids  and  Cones 205 

Spheres 208 

Similar  Solids 209 

Gauging 210 

Synopsis 212 

Metric  System. 213 

Vermont    Partial    Pay- 
ments   227 

Vermont  Taxes.  . . , 231 

Measures  and  Tables 233 


•^¥, 


®P^1^g^^^^5"^^5^c75- 


O  B  AT^    EXERCISES, 

495.  1.  What  is  ^0  of  $100  ?    ^f^?    t¥o?    tVo? 

2.  What  is  yf «  of  $500  ?     Of  $700  ?     Of  $1000  ? 

3.  What  is  ^lo  of  $600?    ^^?    ^?    ^? 

4.  How  many  hundredths  of  $100  are  $5  ?     $7  ?     $18  ? 

5.  How  many  hundredths  of  $500  are  $25  ?  $35  ?  $50  ? 

496.  Percentage  is  a  term  applied  to  computations 
in  which  100  is  employed  as  afxed  measure,  or  standard. 

497.  Per   Cent,  is  an  abbreviation   of   the  Latin 

phrase  per  centum,  which  signifies  by  the  hundred. 

Thus,  5  per  cent,  means  5  of  every  100,  or  yf  ^,  the  5  standing  for 
the  numerator,  and  the  words  ''per  cent.''  for  the  denominator  100. 
Thus,  25  per  cent.  =  ^^^  or  .25. 

498.  The  Sign  of  Per  Cent,  is  %.  It  is  read  per 
cent.     Thus  Q%  is  read  6  per  cent. 


2 


PERCENTAGE. 


What  per  cent,  of  a  number  is  -^  of  it  ?  -^  ?    .08  ? 
.12X?    ^?    ^?    .025?    .OOi?    .04|?    .375?    .0325? 

499.  What  per  cent,  of  a  number  is  ^  of  it  ? 

Ai^ALYSis. — Since  the  whole  of  any  number  is  ^§,  J  of  the 

33- 
same  is  \  of  ^g,  or  :^,  equal  to  33J%.     Hence,  etc. 

What  ^  of  a  number  is  i  of  it ?    }?    \'i    |?    f?    f? 
I?    f?    ^?    if?    V    A?    f*? 

500.  ^hdii  fractional  part  of  a  number  is  12^^  of  it? 

Analysis.— 12^%  is  — ^,  or  ^%^^,  equal  to  ^.    Hence,  etc. 

What  part  of  a  number  is  8^^  of  it?     16|^?     15^? 
20^?    37i^?     7i^?     6i^?    25^?     66f?     75^? 

501.  What  part  of  a  number  is  -J-^  of  it  ? 

1 
Analysis. — \%  is  -^,  equal  to  ^.    Hence,  etc. 
100 

What  is  i^  of  a  number?    \%1    i%?    ^%?    i%? 

503.  Any  per  cent,  may  be  expressed  either  as  a  deci' 
mal  or  as  a  fraction,  as  shown  in  the  following 

Table. 


*er  cent. 

Decimal.          Fraction. 

Per  cent. 

Decimal 

Fraction. 

1^ 

.01            Tftr- 

Wo 

.75 

1 

2% 

.02              ^ 

100^ 

1.00 

^% 

.04             ^ 

125^ 

1.25 

li 

Q% 

.06             ^ 

i% 

.005 

¥^17 

Wo 

.10,  or  .1    r^ 

i% 

.0075 

zis 

Wo 

.20,  or  .2     i 

Wo 

.08i 

^ 

25% 

.25               J 

m% 

.125 

i 

50% 

.50               J 

m% 

.1625 

M 

PERCENTAGE.  3 

WJtITTEN     EXEnCISES, 

603.  Change  to  expressions  having  the  per  cent.  sign. 
1.  .15;  .085;  .33^;  .375;  .00| ;  H;  H;  .75f. 

^.  ^i;  A;  .oof;  h;  I;  iV;  .00125;  |;  2|. 

Change  to  the  form  of  decimals, 

3.  5i^;  9i^;  20^^;  3^^;  i^;  3^^;  If^;  112^^. 
Change  to  the  form  of  fractions, 

4.  24^;  1^;  6^^ ;  37^^;  |^ ;  3^^;  120^;  75^. 

504.  In  the  applications  of  percentage,  at  least  three 
elements  are  considered,  viz. :  the  Rate,  the  Base,  and  the 
Percentage,     Any  two  being  given,  the  other  can  be  found. 

505.  The  Rate  is  the  number  per  cent,  or  the  num- 
ber of  hundredths.     Thus,  in  b%,  .05  is  the  rate.    Hence, 

Bate  per  cent,  is  the  decimal  which  denotes  how  many  hundredths 
of  a  number  are  to  be  taken  or  expressed. 

506.  The  Rase  is  the  number  of  which  the  per  cent, 
is  taken. 

Thus,  in  the  expression,  5^  of  $15,  the  'base  is  $15. 

507.  The  Percentage  is  the  result  obtained  by 
taking  a  certain  per  cent,  of  the  base. 

Thus,  in  the  statement,  6  %  of  $50  is  $3,  the  rate  is  .06,  the  base 
$50,  and  the  'percentage  is  $3. 

508.  The  Amount  is  'the  sum  of  the  base  and  the 
percentage. 

Thus,  if  the  base  is  $80,  and  the  percentage  $5,  the  amount  is 

$80 +  $5  =  $85. 

509.  The  Differ ence  is  the  remainder  found  by 
subtracting  the  percentage  from  the  base. 

Thus,  if  the  base  is  $80,  and  the  percentage  $5,  the  difference  is 

$80  -  $5  =  $75. 


4  PERCENTAGE. 

510.  The  base  and  rate  being  given  to  find  the 
percentage. 


OMAL      EXEItCISES. 


1.  What  is  10%  of  140  ? 

Analysis.— 10^  is  y^^  =  ^,  and  f^  of  140  is  14    Hence  10% 
of  140  is  14. 


What  is 

2.  5^  of  $80? 

3.  7^  of  200  lb.  ? 

4.  6^  of  150  men  ? 

5.  25^  of  120  mi.  ? 

Find  the  amount 

10.  Of  100  A. +27^. 

11.  Of  $75 +  5^. 

12.  Of  32doz.  +  12J^. 


How  much  is 

6.  12^^  of  72  gal.  ? 

7.  40^  of  60  sheep  ? 

8.  S%  of  50  bu.  ? 

9.  50^  of  $240  ? 

Find  the  difference 

13.  Of  90  hhd.  —  10^. 

14.  Of  63  Cd.  -  33^^. 

15.  Of  $200  —  2i^. 


16.  A  farmer  had  150  sheep,  and  sold  20^  of  thera. 
How  many  had  he  left  ? 

17.  A  mechanic  who  received  $20  a  week  had  his  sal- 
ary increased  %%.     What  were  his  daily  wages  then  ? 

18.  From  a  hhd.  of  molasses  containing  63  gal.  33^^ 
was  drawn.     How  many  gallons  remained  ? 

19.  A  grocer  bought  150  dozen  eggs,  and  found  16f^ 
of  them  bad  or  broken.     How  many  were  salable  ? 

20.  A  train  of  cars  running  25  miles  an  hour  increases 
its  speed  12^^.     How  far  does  it  then  run  in  an  hour  ? 

511.  Prikciple. — The  percentage  of  any  number  is 
the  same  part  of  that  iiumher  as  the  given  rate  is  of  100^. 


PERCENTAGE. 


WRITTEN     JEXBRCIS  ES. 

513.  1.  What  is  17^  of  $4957  ? 


OPERATION. 

$4957 


$842.69 


Analysis. — Since  17%  is  .17,  the  required 
percentage  is  .17  of  $4957,  or  $4957  x  .17,  which 
ill         is  $842.69. 


What  is 

2.  35^  of  695  lb.  ? 

3.  75^  of  $8428  ? 

4.  12^^  of  £2105  ? 

Rule. — Multiply  the  iase  iy  the  rate, 
^art  of  the  base  as  the  rate  is  of  100^. 

This  rule  may  be  briefly  expressed  by  the  following 

Formula. — Percentage  =  Base  x  Bate. 


Eind 

5.  33^%  of  8736  bu. 

6.  ^%  of  $35000. 

7.  120^  of  $171.24. 

Or,  take  such  a 


What 

Find 

8.    Is4|^of  312.8rd.? 

13. 

84^  of  354  bu. 

9.     Is  105^  of  $5728? 

14. 

85^  of  -J  of  a  ton. 

10.    Is  $3140.75  +  11^? 

15. 

ifc  of  16400  men. 

11.    Is2|mi.  +  7i^? 

16. 

f  ^  of  1  of  a  year. 

13.    Is  400  ft. -3i^? 

17. 

f  ^  of  if  of  a  hhd. 

18.  The  bread  made  from  a  barrel  of  flour  weighs  35^ 
more  than  the  flour.    What  is  the  weight  of  the  bread? 

19.  A  man  having  a  yearly  income  of  $4550  spends  20;^ 
of  it  the  first  year,  25^  of  it  the  second  year,  and  dll^%  of 
it  the  third  year.     How  much  does  he  save  in  3  years  ? 

20.  A  man  receives  a  salary  of  $1600  a  year.  He  pays 
18^  of  it  for  board,  S^%  for  clothing,  and  16^  for  inci- 
dentals. What  are  his  yearly  expenses,  and  what  does  he 
save  ? 


6  PEECEKTAGE. 

21.  A  man  owning  |^  of  a  cotton-mill,  sold  d6%  of  his 
share  for  $24640.  What  part  of  the  whole  mill  did  he 
still  own,  and  what  was  its  value  ? 

22.  Smith  had  $5420  in  bank.  He  drew  out  16%  of  it, 
then  20^  of  the  remainder,  and  afterward  deposited  12|-^ 
of  what  he  had  drawn.     How  much  had  he  then  in  bank  ? 

513.  The  base  and  percentage  being  given  to  find 
the  rate. 


omatj  exercises. 

1.  What  per  cent,  of  25  is  3  ? 

Analysis— Since  3  is  ^^  of  25, it  is  /^  of  100% ,  or  12^ .    Hence, 

Sis  12%  of  25. 

What  per  cent. 

2.  Of  24  is  18  ? 

3.  Of  $16  are  $4  ? 

4.  Of200  figs  are  20  figs? 

5.  Of  40  lb.  are  15  lb.  ? 

6.  Of  12i  bu.  are  2|^  bu.  ? 

7.  Of  2  A.  are  80  sq.  rd.  ? 

8.  Of  1  da.  are  16  hr.? 

16.  f  of  an  acre  is  what  per  cent,  of  it  ? 

17.  f  of  a  cargo  is  what  per  cent,  of  it  ? 

18.  2^  times  a  number  is  what  per  cent,  of  it  ? 

19.  If  $6  are  paid  for  the  use  of  $30  for  a  year,  what  is 
the  rate  per  cent.  ?  ^  ' 

20.  If  a  milkman  adds  1  pint  of  water  to  every  gallon 
of  milk  he  sells,  what  per  cent,  does  he  add  ? 

514.  Principle. — The  rate  is  the  number  of  hundredths 
which  the  percentage  is  of  the  base. 


What  per  cent. 

'  9.  Are  6^  mi.  of  12imi.? 

10.  Are  18  qt.  of  30  qt.  ? 

11.  Are  16f  cents  of  $1  ? 
^12.  Is  $i  of  $25  ? 

'"l3.  Isf  of  f? 

14.  Isf  of2i? 

15.  Is  I  of  3|? 


PERCENTAGE. 


WBITTEH^    EXEltCISES. 


515.  1.  What  per  cent,  of  72  is  48  ? 

OPERATION. 

72=:.66f  =  66f^ 


48 


Analysis.— Since  the  per- 
centage is  the  product  of 
the  base  and  rate,  the  rate 
Or,   f  I  =  f  ;   100^  X  f  =  ^^%     is  the  quotient  found  by  di- 
viding  the   percentage  by 
the  base ;  and  48  divided  by  72  is  f  f  =  |  =  .66| ;  hence  the  rate  is ' 
66|%.     Or, 

Since  48,  the  percentage,  is  f  of  the  base,  the  rate  is  f  of  100  fo, 
or66|%. 


What  per  cent. 

2.  Of  300  is  75? 

3.  Of  66  is  16i  ? 

4.  Of  $20  are  121.60  ? 


What  per  cent. 

5.  Of  $18  are  90  cents  ? 

6.  Of  560  lb.  are  80  lb.  ? 

7.  Of  980  mi.  are  49  mi.  ? 


EuLE. — Divide  the  percentage  hy  the  base.     Or,  take 
such  a  part  of  100^  as  the  percentage  is  of  the  base. 

Formula. — Bate  —  Percentage  -^  Base. 


*  What  per  cent. 

8.  Of  $480  are  $26.40  ? 

9.  Of  192  A.  are  120  A.  ? 

10.  Of  15  mi.  are  10.99  mi.  ? 

11.  Of  46  gal.  are  5  gal.  3  qt.? 

12.  Of  $4  are  30  cents  ? 

13.  Of  6  bu.  1  pk.  are  4  bu. 

2pk.  6qt.  ? 


What  per  cent, 

14.  Are  448  da. of  5600  da.? 

15.  Are  5  lb.  10  oz.  of  15  lb. 

Avoir.  ? 

16.  Is  13.5  of  225  ? 

17.  Isfiof^^? 

18.  Is  3f  of  181  ? 

19.  Is  22|  of  182.4  ? 


20.  A  grocer  sold  from  a  hogshead  containing  600  lb. 
of  sugar,  \  of  it  at  one  time,  and  \  of  the  remainder  at 
another  time.    What  per  cent,  of  the  whole  remained  ? 

21.  A  merchant  owes  $15120,  and  his  assets  are  $9828. 
What  per  cent,  of  his  debts  can  he  pay  ? 


5  PERCENTAGE. 

516.  Tlie  rate  and  percentage  being  given  to  find 
the  base. 

OMJLIj  jexjemcisjes. 

1.  18  is  3^  of  what  number  ? 

Analysis. — Since  3%,  or  y^^,  of  a  certain  number  is  18,  jj^  is  J 
of  18,  or  6,  and  -JgJ  is  600.    Hence  18  is  3%  of  600. 


Of  what  number 

2.  Is  15  26%  ? 

3.  Is  24  75^  ? 

4.  Is  48  8^? 
6.    Is  1.2  e%? 


Of  what  are 

6.  30  1b.  20^?    25^? 

7.  $84  12^  ?    21^  ? 

8.  15bu.  30^?    50^? 

9.  16Aoz.l2i%?    S^%? 


10.  12^^  of  96  is  33|^  of  what  number  ? 

517.  Principle. — The  base  is  as  many  times  the  per- 
eentage  as  100^  is  times  the  rate. 

WBJTTJEN    XJXEMCISilS* 

518*  1.  144  is  75^  of  what  number  ? 

OPERATION.  Analysis. — Since  the  percent- 

244  -=-  »7»5  =:  192  *^S^  *^  i^e  product  of  the  base  by 

the  rate,  the  base  is  equal  to  the 

Or,   100  -7-  75  =:  4=1^  =:  -|       percentage  divided  by  the  rate ; 

144  X  I  =  192  and  144  ^  .75  is  192.    Or, 

Since  the  rate  is  .75,  the  per- 
eentage  is  j^^^^,  or  f  of  the  base ;  hence  the  base  is  |  of  the  percent- 
age, and  I  of  144  is  192. 

2.  $54  are  15^  of  what  ?       r4.  4.56  A.  are  6%  of  what  ? 

3.  $18.75  are  2^%  of  what  ?  h's.  39.6  lb.  are  1^%  of  what  ? 

EuLE. — Divide  the  percentage  ly  tJie  rate.     Or,  take  a& 
many  times  the  percentage  as  100^  is  times  the  rate. 
Formula. — Base  =  Percentage  -^  Rate. 


PERCENTAGE. 

9 

Of  what  number 

Of  what 

6.     Is  828    120^  ? 

10.     Are  $281.25 

37^^? 

7.     Is  6119  105^^? 

11.     Are  $4578 

84^? 

-  8.     Is  .43-     71f i  ? 

12.     Are  37^  bu. 

6i^? 

9.     Is3H    ^H%? 

13.    Are  1260  bbl. 

IH%? 

- 14.  25%  of  800  bu.  is  2^%  of  how  many  bushels  ? 

15.  A  farmer  sold  3150  bushels  of  grain  and  had  30% 
of  his  entire  crop  left.    What  was  his  entire  crop  ? 

16.  A  man  drew  25%  of  his  bank  deposits,  and  expended 
33^%  of  the  money  thus  drawn  in  the  purchase  of  a  horse 
worth  1250.     How  much  money  had  he  in  bank  at  first  ? 

17.  If  a  man  owning  4:6%  of  a  steamboat  sells  16|^  of 
his  share  for  $5860,  what  is  the  value  of  the  whole  boat  ? 

18.  If  $295,12  are  13^^  of  A's  money,  and  4|^  of  A's 
money  is  8%  of  B's,  how  much  more  money  has  A  than  B  ? 

519.  The  amount,  or  the  difference,  and  the  rate 
being  given  to  find  the  base. 

OHjLZ     jexjemcis  bs. 

1.  What  number  increased  by  25^  of  itself  amounts 
to  60  ? 

Analysis. — Since  60  is  the  number  increased  by  25  %  of  itself, 
it  is  Iff,  or  f  of  the  number  ;  and  if  f  of  the  number  is  60,  the 
number  itself  is  4  times  J  of  60,  or  48. 

2.  What  number  increased  by  8J^  of  itself  is  130  ? 

3.  $70  are  40^  more  than  what  sum  ? 

4.  A  man  sold  a  saddle  for  $18,  which  was  12|-^  more 
than  it  cost  him.     What  did  it  cost  him  ? 

5.  A  grocer  sold  flour  for  $8.40  a  barrel,  which  was  16f ^ 
more  than  he  paid  for  it.     What  did  he  pay  for  it  ? 


10  PERCENTAGE. 

6.  What  number  diminished  by  20^  of  itself  is  40  ? 
Analysis. — Since  40  is  tlie  number  diminished  by  20%  of  itself, 

it  is  -f-^Qy  or  f  of  the  number  ;  and  if  f  of  the  number  is  40,  the 
number  itself  is  5  times  i  of  40,  or  50. 

7.  What  number  diminished  by  6%  of  itself  is  38  ? 

8.  What  sum  diminished  by  50%  of  itself  Is  120.50  ? 

9.  68  yd.  are  15%  less  than  what  number  ? 

10.  A  tailor,  after  using  75%  of  a  piece  of  cloth,  had  9| 
yards  left.     How  many  yards  in  the  whole  piece  ? 

11.  A  sells  tea  at  $.90  a  pound,  which  is  10%  less  than 
he  paid  for  it.     What  did  he  pay  for  it  ? 

WBITTEN     EXERCISES, 

520.  1.  What  sum  increased  by  37%  of  itself  is  $2055? 

OPERATION.  Analysis.— Since 

1 +  .37  =  1.37  the  number  is  in- 

$2055-^1.37z3$1500  ^^^^^^^  ^'^^^^  ^^  ^^ 

.37  of  itself,  $2055 

^^^  is  137%,  or  1.37  the 

Iff  of  $2055  =  $2055-r-137xl00=:$1500  number.      Hence 

$2055    divided    by 

1.37,  is  the  base  or  required  number.     Or, 

Since  $2055,  the  amount,  is  UJ  of  the  base,  100  times  j^y  of 

$2055,  or  $1500,  is  the  base. 

2.  What  number  increased  by  18%  of  itself  equals  2950  ? 

3.  What  sum  increased  by  15%  of  itself  is  $6900? 

4.  What  number  diminished  by  12%  of  itself  is  2640  ? 

OPEKATiON.  Analysis.— Since  the  number 

1  —.12  =3  .88  is  diminished  12%,  or  by  .12  of 

2640  —  88  =  3000  ^*^^^^'  ^^^^  ^^  ^^^*  ^^  '^^  ^^  *^® 
*  number.     Hence  2640  divided  by 

Or,    2640-^22  X  25  =  3000      .88  is  the  base  or  required  num- 

ber.     Or, 

Since  2640,  the  difference,  is  j^o  or  ||  of  the  base,  25  times  -^  of 

2640,  or  3000,  is  the  base. 


PERCENTAGE.  11 

5.  If  the  difference  is  $1000  and  the  rate  20^,  what  is 
the  base  ? 

6.  What  sum  diminished  by  36%  of  itself  equals  $4810  ? 

EuLE. — Divide  the  amount  ly  1  plus  the  rate;    or, 
divide  the  difference  iy  1  r)iinus  the  rate, 

^  D  .  _  i  ^ynount  -~  (1  +  Rate). 

~~   (  Difference  —  (1  —  Rate). 


What  number  increased 

7.  Byl2^ofitself  is3800: 

8.  By  10^  is  39600  ? 

9.  By  15^  is  $2616.25? 
10.  By  22^  is  1098  bu.  ? 


What  number  diminished 

11.  By7i%  of  itself  is  740? 

12.  By  4.%  is  312  acres  ? 

13.  By  8^  is  $2281.60? 

14.  By  37i^  is  $234,625? 


15.  A  man  sold  160  acres  of  land  for  $4563.20,  which 
was  8^  less  than  it  cost.     What  did  it  cost  an  acre  ? 

16.  A  speculator  bought  48  bales  of  cotton,  and  after- 
ward sold  the  whole  for  $2008.80,  losing  7^.  What  was 
the  cost  of  each  bale  ? 

17.  A  dealer  bought  a  quantity  of  grain  by  measure  and 
sold  it  by  weight,  thereby  gaining  1^%  in  the  number  of 
bushels.  He  sold  at  10^  above  the  purchase  price,  and 
received  $4910.976  for  the  grain.     Eequired  the  cost. 

18^  A  merchant,  after  paying  60;^  of  his  debts,  found 
that  $3500  would  discharge  the  remainder.  What  was 
his  whole  indebtedness  ? 

19.  The  net  profits  of  a  mill  in  two  years  were  $6970, 
and  the  profits  the  second  year  were  b%  greater  than  the 
profits  the  first  year.     What  were  the  profits  each  year? 

20.  A  man  sold  two  houses  at  $2500  each  ;  for  one  he' 
received  20^  more  than  its  value  and  for  the  other  20^ 
less.     Eequired  his  loss. 


13  P  E  R  C  E  K  T  A  G  E  . 

^  APPLICATIONS    OF    PERCENTAGE. 

521.  The  applications  of  percentage  are  those  which 
are  independent  of  time,  as,  Profit  and  Loss,  Commission, 
Stocks,  etc.  ;  and  those  in  which  time  is  considered,  as. 
Interest,  Discount,  Exchange,  etc. 

Since  some  one  of  the  four  formulas  of  percentage 
already  considered  will  apply  to  any  of  these  applications, 
the  following  will  serve  as  a  general 

EuLE. — Note  ivTiat  elements  of  Percentage  are  given  in 
the  problem,  a7id  what  element  is  required,  and  then  apply 
the  special  rule  or  formula  for  the  corresponding  case. 

PEOFIT    AI^D    LOSS. 

522.  Profit  and  Loss  are  terms  used  to  express 
the  gain  or  loss  in  business  transactions. 

523.  Gains  and  losses  are  usually  estimated  at  a  7*ate 
per  cent,  on  the  cost,  or  the  money  or  capital  invested. 

524.  The  operations  involve  the  same  principles  as 
those  of  Percentage. 

525.  The  corresponding  terms  are  the  following  : 

1.  The  Base  is  the  Cost,  or  capital  invested. 

2.  The  Mate  is  the  per  cent,  of  profit  or  loss. 

3.  The  Percentage  is  prQfit  or  loss. 

4.  The  Amount  is  the  cost  p)lus  the  profit,  or  the 
Selling  Price. 

5.  The  Difference  is  the  cost  minus  the  loss,  or  the 
Selling  Price, 


PRO  FITAKD     LOSS.  13 

OnAL     BXERCISES. 

536.  1.  A  horse  that  cost  $200  was  sold  at  a  gain  of 

12^.     What  was  the  gain,  and  the  selling  price  ? 

Analysis. — Since  the  gain  was  12  ^ ,  it  was  -^^^  of  $200,  which  is 
$24  ;  and  the  selling  price  was  $200  +  $24 = $224.  Hence,  etc.  (510.) 

2.  A  saddle  that  cost  $25  sold  at  a  loss  of  10^.  What 
was  the  loss,  and  the  selling  price  ? 

3.  A  tailor  bought  cloth  at  $6  a  yard,  and  wished  to 
sell  it  at  a  gain  of  25^.     At  what  price  must  he  sell  it  ? 

4.  For  how  much  must  a  grocer  sell  tea  that  cost  $.60 
a  pound,  to  gain  30^  ? 

5.  A  merchant  buys  gloves  at  $.75  a  pair,  and  sells  them 
at  a  profit  of  33 J^;^.     For  how  much  does  he  sell  them  ? 

6.  Bought  a  carriage  for  $160,  and,  after  paying  10^ 
for  repairs,  sold  it  at  12^^  profit.  What  was  the  gain, 
and  the  selling  price  ? 

7.  If  butter  bought  at  36  cents  a  pound  is  sold  at  a  loss 
of  16f^,  what  is  the  selling  price? 

8.  What  must  be  the  selling  price  of  coffee  that  cost 
25  cents  a  pound,  in  order  to  gain  20^? 

9.  At  what  price  must  an  article  that  cost  $5  be  sold, 
to  gain  100^?     120^?     150^?    200^? 

537.  1.  A  merchant  bought  cloth  at  $5  a  yard,  and 

sold  it  at  $6  a  yard.     What  was  the  gain  per  cent.  ? 

Analysis. — The  whole  gain  is  the  difference  between  $6  and  $5, 
which  is  $1.  Since  $5  gain  $1,  or  |  of  itself,  the  gain  per  cent,  is 
J  of  100%  or  20%.     Hence,  etc.     (513.) 

2.  What  is  gained  per  cent,  by  selling  coal  at  $7  a  tott, 
that  cost  $6  a  ton  ? 

3.  Sold  a  piano  for  $300.  which  was  f  of  what  it  cost. 
What  was  the  loss  per  cent  ? 


14:  PERCENTAGE. 

4.  Sold  melons  for  $.75  that  cost  $.50.  What  was  the 
gain  per  cent.  ? 

5.  What  is  gained  per  cent,  by  selling  pine-apples  at  30 
cents  each,  that  cost  $15  a  hundred  ? 

6.  Sold  a  sewing  machine  at  a  loss  of  -J  of  what  it  cost. 
What  was  the  loss  per  cent.  ? 

7.  What  %  is  gained  on  goods  sold  at  double  the  cost  ? 

8.  What  %  is  lost  on  goods  sold. at  one-Jialf  i\\Q  cost? 

9.  What  per  cent,  profit  does  a  grocer  make  who  buys 
sugar  at  10  cents  and  sells  it  at  12  cents  ? 

10.  What  per  cent,  is  gained  on  an  article  bought  at  $3 
and  sold  at  $5  ? 

■*     538.  1.  A  dealer  sold  flour  at  a  profit  of  $2  a  barrel, 
and  gained  25^.     What  was  the  cost  ? 

Analysis.— Since  the  gain  was  25%  =  ^^^,  or  J,  $2  is  J  of  the 
cost ;  $2  is  i  of  4  times  $2,  or  $8.     Hence,  etc.    (516.) 

3.  Sold  hats  for  $1  less  than  cost,  and  lost  16f  ;^.  What 
did  they  cost  ? 

3.  A  merchant  sells  silk  at  a  profit  of  $1 1^  a  yard,  which 
is  40^  gain.  What  did  it  cost,  and  what  is  the  selling 
price  ? 

4.  If  com  selling  for  21  cents  a  bushel  more  than  cost 
gives  a  profit  of  30^,  what  did  it  cost  ? 

5.  Sold  sheep  at  $2^  more  than  cost,  which  was  a  profit 
of  50^.    What  did  they  cost,  and  what  is  the  selling  price  ? 

6.  Shoes  sold  at  $.50  above  cost  giye  a  profit  of  12|^^. 
What  did  they  cost  ? 

7.  A  farmer,  by  selling  a  cow  for  $12  less  than  she 
cost,  lost  33^^.  '  What  did  she  cost  ? 

8.  A  grocer  sells  a  certain  kind  of  tea  for  6  cents  a 
pound  more  than  cost  and  gains  6%,    What  did  it  cost  ? 


PEOFITAi^DLOSS.  15 

539.  1.  A  watch  was  sold  for  $120,  at  a  gain  of  20^. 
What  was  the  cost  ? 

Analysis.— Since  the  gain  was  20%,  or  i,  of  the  cost,  $120,  the 
selling  price,  is  |  of  the  cost.  J  of  $120,  or  $20,  is  J  of  the  cost,  and 
I,  or  the  cost  itself,  is  5  times  $20,  or  $100.    Hence,  etc.    (518.) 

2.  Sold  tea  at  $.  90  a  pound,  and  gained  26%.  What 
did  it  cost  ? 

3.  A  newsboy,  by  selling  his  papers  at  4  cents  each, 
gains  33^%,     What  do  they  cost  him  ? 

4.  A  man  sold  a  horse  and  harness  for  $330,  which  was 
10^  more  than  they  cost.     What  was  their  cost  ? 

5.  If  20^  is  lost  by  selling  wheat  at  $1.60  a  bushel, 
what  would  be  gained  if  sold  at  20^  above  cost  ? 

6.  John  Eice  lost  40^  on  a  reaper,  by  selling  it  for  $60. 
For  what  should  he  have  sold  it  to  gain  40^  ? 

7.  If,  by  selling  books  at  $2  a  volume,  there  is  a  gain 
of  25%,  at  what  price  must  they  be  sold  to  lose  15^? 

8.  Two  pictures  were  sold  for  $99  each  ;  on  one  there 
was  a  gain  of  10^,  on  the  other  a  loss  of  10^.  Was  there 
a  gain  or  loss  on  the  sale  of  both,  and  how  much  ? 

^  WRITTEN     EXJEHCISJES. 

530.  1.  A  hogshead  of  sugar  bought  for  $108.80  was 
sold  at  a  profit  of  12^^.     What  was  the  gain  ? 

OPERA.TION.— $108.80  X  .12i  =  $13.60.     (512.) 
Formula. — Profit  or  Loss  =  Cost  x  Bate  %. 

Find  the  Profit  or  Loss, 

2.  On  land  that  cost  $1745,  and  was  sold  at  a  gain  of  20^. 

3.  On  goods  that  cost  $3120,  and  were  sold  at  27'<^  gain. 

4.  On  a  boat  bought  for  $2545|^,  and  sold  at  25^  loss. 


16  PERCENTAGE. 

5.  On  goods  bought  for  $2560.75,  and  sold  at  S%  loss. 

6.  On  25  tons  of  iron  rails  bought  at  %bS  a  ton,  and 
sold  at  an  advance  of  11  \%. 

7.  A  merchant  pays  $6840  for  a  stock  of  spring  goods, 
and  sells  them  at  an  advance  of  26^^  on  the  purchase 
price.    After  deducting  $3  75  for  expenses,  what  is  his  gain  ? 

-  8.  A  miller  bought  1000  bushels  of  wheat  at  $1.84  a 
bushel,  and  sold  the  flour  at  16f^  advance  on  the  cost  of 
the  wheat.     What  was  his  profit  ? 

9.  Bought  128  tons  of  coal  at  %b,l^  a  ton,  and  sold  it 
at  a  gain  of  22^.    What  was  the  entire  profit  ? 

10.  A  ship,  loaded  with  3840  bbl.  of  flour,  being  over- 
taken by  a  storm,  found  it  necessary  to  throw  37^^  of  her 
cargo  overboard.     What  was  the  loss  at  $7.65  a  bbl.  ? 

11.  A  man  bought  a  pair  of  horses  for  $450,  which  was 
25^  less  than  their  real  value,  and  sold  them  for  25^  more 
than  their  real  value  ;  what  was  his  gain  ? 

531.  1.  Bought  a  house  for  $4380.     For  what  must  it 
be  sold  to  gain  14|^? 
Operation.— $4380  x  (1  +  .14^)  or  1.145  =  $5015.10.    (512.) 

2.  At  what  price  must  pork,  bought  at  $18.40  a  barrel, 
be  sold,  to  lose  15^? 

Operation.— $18.40  x  (1  -  .15),  or  .85  =  $15.64.    (512.) 

^  CI  IT      -n  •  (  Cost  X  (1  +  Rate  %  of  Gain). 

FoKMULA.—Sellmg  Prices  ]  ^    ^     /-,-r»x^i.-r      ^ 
^  ^^  (  Cost  X  (1— Eate  %  of  Loss). 

Find  the  Selling  Price, 

3.  Of  goods  bought  at  $187.50,  and  sold  at  11^^  gain. 

4.  Of  beef  bought  at  $12|  a  barrel,  and  sold  at  9^^  loss. 

5.  Of  cotton  bought  at  $.14,  and  sold  at  a  gain  of  21|^. 
o.  Of  cloth  that  cost  $5^  a  yard,  and  was  sold  at  a 

profit  of  18^^  ? 


PROFIT     AND     LOSS.  17 

7.  At  what  price  must  goods  that  cost  $3^  a  yard  be 
marked,  to  gain  26%  ?    To  lose  20^  ? 

8.  Sold  a  lot  of  damaged  goods  at  a  loss  of  16%.  What 
was  the  selling  price  of  those  that  cost  $.62^  ?     $1.25  ? 

9.  Bought  a  hogshead  of  sugar  containing  9  cwt.  56  lb. 
for  $86.04,  and  paid  $4.78  freight  and  cartage.  At  what 
price  per  pound  must  it  be  sold  to  gain  20^  ? 

^   533.  1.  Bought  wool  at  $.48  a  pound,  and  sold  it  at 
$.  60  a  pound.     What  per  c§nt.  was  gained  ? 

Operation.— $.60  -  $.48  =  $.  12 ;  and  $.12  -f-  $.48  =r  .25  =  25% . 
(515.) 

2.  Sold  for  $10.02  an  article  that  cost  $12.  What  was 
the  loss  per  cent.  ? 

Operation.— $12-$10.02=:$1.98;  and$l  98^$12=.16|=16i%. 
Formula.— i^^^fe  %  =.  Profit  or  Loss  -r-  Cost. 
^  Find  the  rate  per  cent,  of  profit  or  loss, 

3.  On  sugar  bought  at  8  cents  and  sold  at  9^  cents. 

4.  On  tea  bought  at  $1,  and  sold  at  $.87|-. 

5.  On  goods  that  cost  $275,  and  were  sold  for  $330. 

6.  On  grain  bought  for  $1.25  a  bushel,  and  sold  for 
$1.60  a  bushel. 

7.  On  a  sewing-machine  sold  for  $72.96,  at  again  of 
$9.12. 

8.  On  goods  sold  for  $425.98,  at  a  loss  of  $134.52. 

9.  Bought  paper  at  $3  a  ream,  and  sold  it  at  25  cents 
a  quire.    What  was  the  gain  per  cent.? 

10.  A  dealer  bought  108  bbl.  of  apples  at  $4.62^,  and 
sold  them  so  as  to  gain  $114. 88|.     What  was  his  gain  ^? 

11.  If  1^  of  an  acre  of  land  is  sold  for  f  the  cost  of  an 
acre,  what  is  the  gain  per  cent.  ? 


18  PERCEKTAGE. 

12.  If  f  of  an  acre  of  land  is  sold  for  the  cost  of  ^  of 
an  acre,  what  is  the  loss  per  cent.  ? 

13.  If  I  of  a  chest  of  tea  is  sold  for  what  the  whole 
chest  cost,  what  is  the  gain  per  cent,  on  the  part  sold  ? 

533.  1.  A  speculator  sold  grain  at  a  profit  of  33^%,  by 
which  he  made  25  cents  on  a  bushel.    What  did  it  cost  ? 

Operation.— $.25-5-.33i=$.75.    Or,  $.25-4-1= $.75.    (518.) 

2.  Lost  $45. 75  on  the  sale  of  a  horse,  which  was  20% 
of  the  cost.    What  was  the  cost  ? 

Operation.— $45.75h-.20=$228. 75.    Or  $45.75 -5- J =$228. 75. 

Formula. — Cost  =  Profit  or  Loss  -f-  Bate  %. 

Find  the  Cost, 

3.  Of  goods  sold  at  $1500  profit,  or  a  gain  of  16^. 

4  Of  fiour  sold  at  a  loss  of  $.88,  or  10^,  on  a  barrel. 

5.  Of  wheat  sold  at  a  loss  of  6  cents,  or  4^,  on  a  bu.  ? 

6.  Of  lumber  sold  at  an  advance  of  $4.95  per  M.,  or 
35^  gain. 

7.  If  a  grocer  sells  his  stock  at  a  profit  of  15^,  what 
amount  must  he  sell  to  clear  $2500  ? 

8.  A  and  B  engage  in  speculation.  A  gains  $2000, 
which  is  12|^^  of  his  capital,  and  B  loses  $500,  which  is 
6%  of  his  capital.    What  sum  did  each  invest? 

534.  .1.  A  furniture  dealer  sold  two  parlor  sets  for 

$450  each  ;  on  one  he  made  15^,  on  the  other  he  lost  15^. 

What  did  each  cost  him  ? 

Operation       ($450^(1 +  .15)=$391. 30  +  ,  cost  of  one. 
UPERATION.      j  $45o^^i_.i5)::,|529.41  +  ,  costof  theother.  (520.) 

T?  ^^  4     cr77-"'T>-     .    {{^  +  R(^ie%  of  gain.) 

■FouMVLA.-Cost=Selhng  Price-^  |  ^i_^,,^^f  fo,,.) 


PROFIT     AKD     LOSS.  19 

Find  the  Cost^ 

2.  Of  coal  sold  at  $6,  being  at  a  loss  of  12J^. 

3.  Of  grain  sold  at  1.96  a  bushel,  at  a  gain  of  28%. 
L  Of  silk  sold  for  $5.40  a  yard,  at  a  profit  of  10^. 

5.  Of  hops  sold  at  16  cents  a  pound,  at  a  loss  of  20^. 

6.  Of  fruit  sold  for  $207.48,  at  a  loss  of  lb%. 

,^  7.  Having  used  a  carriage  1  year,  I  sold  it  for  $125, 
which  was  25^  below  cost.  What  should  I  have  received 
had  I  sold  it  for  10^  above  cost  ? 

8.  B  sold  a  span  of  horses  to  C  and  gained  12|^^ ;  0 
sold  them  to  D  for  $550,  and  lost  16f  ^.  What  did  the 
horses  cost  B  ? 

9.  If  a  piece  of  property  increases  in  value  each  year  at 
the  rate  of  25^  on  the  value  of  the  previous  year,  for  4 
years,  and  then  is  worth  $16000,  what  did  it  cost  ? 

535.  1.  Bought  cloth  at  $3.60  a  yard.  At  what  price 
must  it  be  marked  that  12|^  may  be  abated  from  the 
asking  price,  and  still  a  profit  made  of  16f  ^  ?\. 

Operation  ~  \  ^^^^""^ ^^^'^^    =^^-^^  ^  (1  +  •l6|)-$4.20. 

(if«r^^/^5rPnce=$4.30-^(l-.12l)=:$4.80.    (519.) 

2.  At  what  price  must  shovels  that  cost  $1.12  each  be 
marked  in  order  to  abate  h%,  and  yet  make  25^^  profit  ? 

3.  How  must  a  watch  be  marked,  that  cost  $120,  so 
that  4^  may  be  deducted  and  a  profit  of  20^  be  made  ? 

4.  A  merchant,  on  opening  a  case  of  goods  that  cost 
$.80  a  yard,  finds  them  slightly  damaged.  How  must  he 
mark  them,  to  fall  25^  in  his  asking  price,  and  sell  at  cost? 

5.  Bought  land  at  $60  an  acre  ;  how  much  must  I  ask 
an  acre,  that  I  may  deduct  25^  from  my  asking  price,  and 
still  make  20^  on  the  purchase  price  ? 


20  PEBCEKTAGE. 

COMMISSIOK 

536.  An  Agent  or    Commission    Mer'chant 

is  a  person  who  buys  or  sells  merchandise^  or  transacts 
other  business  for  another,  called  the  Pmicipal 

537.  Commission  is  the  fee,  or  compensation, 
allowed  an  agent  or  commission  merchant  for  transacting 
business,  and  is  usually  computed  at  a  certain  rate  per 
cent,  of  the  money  involved  in  the  transaction. 

538.  A  Consignment  is  a  quantity  of  goods  sent 
to  a  commission  merchant  to  be  sold. 

539.  The  Consignor  is  the  person  who  sends  the 
goods  for  sale.    A  consignor  is  sometimes  called  a  Shipper. 

540.  The  Consignee  is  the  person  to  whom  the 
goods  are  sent.     He  is  sometimes  called  a  Correspondent. 

541.  The  Net  Proceeds  of  a  sale  or  other  transac- 
tion is  the  sum  of  money  that  remains  after  all  expenses 
of  commission,  etc.,  are  paid. 

543.  A  Guaranty  is  security  given  by  a  commis- 
sion merchant  to  his  principal  for  the  payment  of  goods 
sold  by  him  on  credit. 

543.  An  Account  Sales  is  a  written  statement 
made  by  a  commission  merchant  to  his  principal,  contain- 
ing an  account  of  goods  sold,  their  price,  the  expenses^ 
and  the  net  proceeds. 

544.  A  JBroker  is  a  person  who  buys  or  sell  stocks, 
bills  of  exchange,  real  estate,  etc.,  for  a  commission, 
which  is  called  BroJcerage. 


COMMISSION.  21 

545*  The  principles  and  operations  of  Percentage  in- 
volved in  Commission  and  Brokerage  are  the  same  as 
those  already  treated. 

54:6.  The  following  are  the  corresponding  terms  : 

1.  The  Base  is  the  amount  of  sales,  money  invested, 
or  collected. 

2.  The  Rate  is  the  per  cent,  allowed  for  services. 

3.  The  JPercentage  is  the  Commission  or  Broker- 
age. 

4.  The  Amount  or  Difference  is  the  amount  of 
sales,  plus  or  minus  the  commission. 

WRITTEN    EXERCISES. 

547.  Find  the  Commission  or  BroTcerage^ 

1.  On  a  sale  of  flour  for  $2575,  at  %\%. 

Operation.— $2575  x  .025  =  $64.37i     (512.) 

Formula. — Amount  of  Sales  x  Rate  %  =  Commission. 

2.  On  the  purchase  of  a  farm  for  $13750,  at  2|^. 

3.  On  the  sale  of  a  mill  for  $9384,  at  |^. 

4.  On  the  sale  of  $21680  worth  of  wool,  at  1^%. 

5.  On  the  sale  of  250  bales  of  cotton,  averaging  520  lb., 
at  14|  cents  a  pound  ;  commission  1^%. 

6.  On  the  sale  of  175  shares  of  stock,  at  $92|  a  share  ; 
brokerage,  ^%. 

7.  On  the  sale  at  auction  of  a  house  and  the  furniture 
for  $9346.80,  at  6}^. 

8.  A  commission  merchant  sells  225  bbl.  of  potatoes 
at  $3.25  per  bbl.,  and  316  bbl.  of  apples  at  $4^  per  bbl. 
What  is  his  commission  at  4^^  ? 


22  PERCE2!^TAGE. 

548.  Find  the  rate  of  commission  or  brokeiBge, 

1.  When  $89  commission  is  paid  for  selling  goods  for 
$3560. 

Operation.— 89  -^  3560  =:  m\  =  2J-^.    (515.) 

Formula. — Com7nission  -^  A^nount  of  Sales  =  Rate  %. 

2.  When  $165  com.  is  paid  for  selling  goods  for  $4950. 

3.  When  $63  is  paid  for  collecting  a  debt  of  $1260. 

4.  When  $117.75  is  paid  for  selling  a  house  for  $7850. 

5.  When  1235.40  is  paid  for  buying  26750  lb.  of  wool 
at  32  cents  a  pound. 

6.  When  $125  is  paid  for  the  guaranty  and  sale  of  goods 
for  $2500. 

7.  Paid  my  N.  0.  agent  $74.25  for  buying  26400  lb.  of 
rice,  at  4|^  ct.  a  lb.    What  was  the  rate  of  his  commission  ? 

549.  Find  the  Amount  of  Sales, 

1.  Wlien  a  commission  of  $147  is  charged  at  Z^%. 
Operation.— $147  -^  .035  =  $4200.    (517.) 

Formula. — Commission  -r-  Rate  %  =  Amount  of  Sales. 

2.  When  $92.80  commission  is  paid  at  3^%. 

3.  When  $210  commission  is  charged  at  6%. 

4.  When  $24  brokerage  is  paid  at  i%. 

5.  When  $135  commission  is  charged  at  1^%. 

6.  Paid  an  attorney  $72.03  for  collecting  a  note,  which 
was  a  commission  of  '7^%.    What  was  the  face  of  the  note  ? 

"^   550.  Find  the  Amount  of  Sales, 

1.  When  the  net  proceeds  are  $4875,  commission  2^%. 
Operation.— $4875  ^  .975  =  $5000.    (519.) 

Formula. — Net  proceeds -^-{1—-  Rate  %)=zAmt.  of  Sales. 

2,  When  the  net  proceeds  are  $3281.25,  commission  12^^. 


COMMISSION.  23 

3.  When  the  net  proceeds  are  $560,  and  the  com.  4^. 

4.  After  deducting  6^%  commission  and  $132  for 
storage,  my  correspondent  sends  me  $23654.25  as  the  net 
proceeds  of  a  consignment  of  pork  and  flour.  What  was 
the  gross  amount  of  the  sale  ? 

551.  Find  the  amount  to  be  invested , 

1.  If  $9500  is  remitted  to  a  correspondent  to  be  invest- 
ed in  woolen  goods,  after  deducting  6%  commission. 

Operation.-  $9500  -f- 1.05  ==  $9047.62.    (519.) 

¥oRM.VLA,— Amount  Remitted-^  (1  +  Rate  %)  ■=  Sum 
Invested. 

2.  If  $4908  be  remitted,  deducting  ^\%  commission. 

3.  If  $3246.20  be  remitted,  deducting  %%  commission. 

4.  If  $1511.25  be  remitted,  deducting  ^%  commission. 

5.  If  $10701.24  be  remitted,  deducting  ^%  brokerage. 

6.  A  dealer  sends  his  agent  in  Havana  $6720.80,  with 
which  to  purchase  oranges  and  other  fruits,  after  deduct- 
ing his  commission  of  b%.  What  sum  did  the  agent  invest, 
and  what  was  the  amount  of  his  commission  ? 

^  7.  What  amount  of  sugar  can  be  bought  at  8  cents  a 
pound,  for  $2523.40,  ?ifter  deducting  a  commission  of  1^%, 

8.  Eemitted  to  a  stockbroker  $10650,  to  be  invested  in 
stocks,  after  deducting  \%  brokerage.     What  amount  of 

.  stock  did  he  purchase  ? 

9.  A  broker  received  $45337.50  to  invest  in  bond  and 
mortgage,  after  deducting  a  commission  of  ^\%,  What 
amount  did  he  invest,  and  what  was  his  commission  ? 

^  10.  Sent  $250.92  to  my  agent  in  Boston,  to  be  invested 
in  prints  at  15  cents  a  yard,  after  taking  out  his  commis- 
sion of  ^%,     How  many  yards  ought  I  to  receive  ? 


24  PERCEII^TAGE. 

REVIEW. 
OMAZ      EXEHCISES. 

553.  1.  If  stoves  bought  at  $36  each  are  sold  at  a 
profit  of  ^\%,  what  is  the  gain? 

2.  What  will  be  the  expense  of  collecting  a  tax  of  $1000, 
allowing  b%  ? 

3.  What  will  a  broker  receive  for  selling  $600  worth  of 
stock,  at  ^%  brokerage  ? 

4.  A  man  having  $250  spent  $80.  What  per  cent,  of 
nis  money  had  he  left  ? 

5.  If  a  man  sells  a  building  lot  that  cost  $300,  at  an 
advance  of  166f^,  what  is  his  gain  ? 

6.  I  of  30^  is  what  per  cent,  of  72^  ?     Of  144^?    Ofv 
180^?     240^? 

7.  Bought  a  horse  for  20^  less  than  $200,  and  sold  him 
for  10^  more  than  $200.    What  per  cent,  was  gained  ? 

8.  How  many  bushels  of  wheat  at  $2  a  bushel  can  an 
agent  buy  for  $2040,  and  retain  2%  on  what  he  expends 
as  his  commission  ? 

9.  If  by  selling  land  at  $150  an  acre  I  lose  25^,  how 
must  I  sell  it  to  gain  40^  ? 

10.  A  boy  bought  bananas  for  $3  a  hundred,  and  sold 
them  for  5  cents  each.     What  per  cent,  did  he  gain  ? 

11.  Bought  cannel  coal  at  $19  a  ton,  which  was  ^%  less 
than  the  market  price.     What  was  the  market  price  ? 

12.  Paid  an  agent  $150,  or  a  commission  of  1\%,  for 
selling  my  house.     For  what  sum  was  the  house  sold  ? 

13.  If  an  article  is  sold  so  as  to  gain  f  as  much  as  it 
cost,  what  per  cent,  is  gained  ? 


REVIEW.  25 

14.  A  merchant  tailor  sold  some  linen  coats  at  $1.80 
each,  which  was  33^%  below  the  marked  price.  What 
was  the  marked  price  ? 

15.  A  grocer  bought  40  gal.  of  maple  syrup  at  the  rate 
of  4  gal.  for  $6,  and  sold  it  at  the  rate  of  5  gal.  for  $8. 
What  was  his  whole  gain,  and  his  gain  per  cent.  ? 

16.  How  much  wheat  must  a  farmer  take  to  mill  that 
he  may  bring  away  the  flour  of  4|  bushels,  after  the  miller 
takes  his  toll  of  10^? 

WRITTEN     EXERCISES. 

553.  1.  After  taking  out  15^  of  the  grain  in  a  bin, 
there  remained  40  bu.  3|  pk.  How  many  bushels  were 
tl^e  at  first  ? 

/^2.  The  net  profits  of  a  farm  in  2  years  were  $3485,  and 

,,  the  profits  the  second  year  were  b%  greater  than  the 

profits  the  first  year.     What  were  the  profits  each  year  ? 

3.  A  has  32^  more  money  than  B  ;  what  per  cent,  less 
is  B's  money  than  A's  ? 

4.  Bought  450  bushels  of  wheat  at  $1.25  per  bushel,  and 
sold  it  at  $1.40  per  bushel.  What  was  the  whole  gain, 
and  the  gain  per  cent.  ? 

5.  A  man  drew  out  of  the  bank  |  of  his  money,  and  ex< 
pended  30^^  of  50^  of  this  for  728  bu.  of  wheat,  at  $1.12^ 
a  bushel.     What  sum  had  he  left  in  bank  ? 

6.  Sold  goods  to  the  amount  of  $47649,  at  a  profit  of 
16f^.     Eequired  the  cost  and  the  total  gain. 

7.  A  broker  received  $37.50  for  selling  some  uncurrent 
money,  charging  \%  brokerage.     How  much  did  he  sell  ? 

8.  If  f  of  a  farm  is  sold  for  what  |  of  it  cost,  what  is 
the  gain  per  cent.  ? 


26  PERCEKTAGE. 

9.  An  architect  charged  ^%  for  plans  and  specifications, 
and  If ^  for  superintending  a  building  that  cost  $25000. 
What  was  the  amount  of  his  fee  ? 

10.  If  a  stationer  marks  his  goods  50^  above  cost,  and 
then  deducts  50^,  what  per  cent,  does  he  make  or  lose? 

11.  Sold  a  farm  for  $14700,  and  lost  12^.  What  per 
cent,  should  I  have  gained  by  selling  it  for  $21000  ? 

12.  If  an  article  bought  at  20^  below  the  asking  price 
is  sold  at  16%  below  that  price,  what  is  the  rate  of  gain  ? 

13.  A  commission  merchant  sold  a  consignment  of 
goods  for  $5250,  and  charged  3^%  commission,  and  2j^% 
for  a  guaranty.    Find  the  net  proceeds. 

14.  Smith  &  Jones  bought  a  stock  of  groceries  for 
$13680.^  They  sold  ^  of  the  entire  stock  at  15^  profit,  ^ 
at  18|-^,  ^  at  20^,  and  the  remainder  at  33^^  profit.  What 
was  the  whole  gain,  and  the  average  gain  per  cent.  ? 

15.  Give  the  marking  prices  at  25^  advance,  of  the 
following  bill  of  goods,  and  the  amount  when  sold  at  a 
reduction  of  10^  from  those  prices : 

1  Case  of  Prints,  450  yd.,  @  $.12 

3  Pieces  Cassimeres,  65    ''    %  3.25 

1  Bale  Ticking,  244   ''    ®    .20 

25  Dress  Shawls,  @  7.36 

1  Gr.  gross  Clark's  Thread,  144  doz.,  @    .70 

50  Gross  Buttons,  @  1 .  00 

16.  How  much  would  the  above  bill  of  goods  amoun; 
to  if  sold  at  5|^^  below  a  marking  price  of  15^  above  cost  ? 

17.  What  would  be  the  net  proceeds  of  a  sale  of  18  cwt. 
75  lb.  of  sugar,  at  $9f  per  cwt.,  allowing  2^;^  commission, 
and  $1,6|  for  other  charges? 


COMMISSION.  27 

18.  A  broker  receives  $7125  to  invest  in  cotton,  at  11 J 
cents  a  pound.  If  his  commission  is  2^%,  how  many 
pounds  of  cotton  can  he  buy? 

19.  If  the  sale  of  potatoes  at  $.75  a  barrel  above  cost 
gives  a  profit  of  18|^,  how  much  must  be  added  to  this 
price  to  realize  a  profit  of  31^%  ? 

20.  An  agent  in  Chicago  purchases  1000  bbl.  of  flour 
at  $6. 80,  and  pays  5  cents  a  barrel  storage  for  30  days ; 
also,  3000  bu.  of  wheat  at  $1.20.  He  charges  a  commis- 
sion of  1^%  on  the  flour,  and  1  cent  a  bushel  on  the  wheat. 
What  sum  of  money  will  balance  the  account,  and  what  is 
the  amount  of  his  commission  ? 

21.  An  agent  in  Boston  received  28000  lb.  of  Texas 
cotton,  which  he  sold  at  $.12|^  a  pound.  He  paid  $45.86 
freight  and  cartage,  and  after  retaining  his  commission, 
he  remits  his  principal  $3252.89  as  the  net  proceeds  of  the 
gale.     What  was  the  rate  of  his  commission? 

22.  The  following  bill  of  goods  was  sold  at  auction  : 
IJ  bbl.  A  Sugar,  312  lb.,  @  $.12|^  that  cost  $.11^ 

I   ''    Pulv.  ''  96  ''    %    .\\\    "       "      .14 

1  Chest  Y.  H.  Tea,       84^^    @  1.10      "      ''    1.12^ 
1  Box  Soap,  60  "    %    .13      "       "      .10| 

1^  Sacks  Java  Coffee,  110  ^^    %    .22^    "      "     .24| 
184  lb.  Codfish,  @    .07i    "      "      .08| 

Allowing  a  commission  of  4^^  for  selling,  find  the  entire 
profit  or  loss,  and  the  gain  or  loss  per  cent,  on  the  whole. 

23.  A  merchant  in  New  York  imported  2400  yd.  of 
English  cloth,  for  which  he  paid  in  London  10s.  sterling 
a  yard,  and  the  total  expenses  were  $255.  He  sold  the 
cloth  for  $3.81  a  yard,  U.  S.  money.  What  was  his  whole 
gain,   and  his  gain  per  cent.  ? 


28 


PERCENTAGE. 


554. 


SYNOPbiS  FOR  EEVIEW. 


1.  Definitions. 


2.  Elements. 


''  1.  Percentage.  2.  Per  Cent.  3.  Sign 
of  Per  Cent.  4.  Rate,  or  Rate  ^ . 
5.  Base.  6.  Percentage.  7.  Amount. 
8.  Difference. 

(  1.  How  many  considered. 
(  2.  How  many  must  be  given. 


3.  510.  1.  Principle.  2.  Rule.  3.  Formula. 

4.  513.  1.  Principle.  2.  Rule.  3.  Formula. 

5.  516.  1.  Principle.  2.  Rule.  3.  Formula. 

6.  519.  1.  Principle.  2.  Rule.  3.  Formula. 


Afplications    of 
Percentage. 


8.  Profit  and  Loss. 


9.  Commission. 


(  1.  Diff't  kinds.  -1  ^-   "^^'^^'^^^  '^'^' 
\  I  2.   With  Time. 

(  2.  General  Rule. 

C  1.  Definition. 

J  2.  To  estimate  gains  and  losses. 
1.  Base. 


3 


Correspond- 
ing terms. 


1.  Definitions.   < 


2.  Rate. 

3.  Percentage. 

4.  Am't  and  Biff. 

1.  Agent,  or  Com- 
mission  Merchant, 

2.  Commission. 

3.  Consignment. 

4.  Consignor. 

5.  Consignee. 

6.  Net  Proceeds. 

7.  Guaranty. 

8.  Account  Sales. 

9.  Broker. 

2.  Prin.  and  Operations  Involved. 

1.  Base. 

2.  Rate. 

3.  Percentage. 

4.  Am't  and  Biff, 


3.  Correspond- 
ing terms. 


OUAZ,    EXERCISES, 

555.  1.  When  h%  is  charged  for  the  use  of  money, 
how  many  dollars  should  be  paid  for  the  use  of  $100  ? 
For  the  use  of  $200  ?    Of  1500  ?     Of  $50  ? 

2.  At  1%  a  year,  what  should  be  paid  for  the  use  of 
$100  for  2  years  ?     Of  $200  for  3  years  ? 

3.  If  $500  is  loaned  for  3  years,  what  should  be  paid 
for  its  use,  at  b%  a  year  ?    At  6^  a  year  ? 

4.  If  I  borrow  $250,  and  agree  to  pay  ^%  a  year  for  its 
use,  how  much  will  be  due  the  lender  in  5  years  ? 

5.  If  $7  is  paid  for  the  use  of  $100  for  1  year,  what  is 
the  per  cent,  ? 

6.  If  $50  is  paid  for  the  use  of  $100  for  5  years,  what 
is  the  per  cent.  ? 

7.  If  $14  is  paid  for  the  use  of  $200  for  1  year,  what  is 
the  per  cent.  ? 

8.  At  6^,  what  decimal  part  of  the  money  borrowed  is 
equal  to  the  money  paid  for  its  use  ?   At  7^  ?   8^  ?   9^  ? 

DEFINITIONS. 

556.  Interest  is  a  sum  paid  for  the  use  of  money. 

557.  The  JPrincipal  is  the  sum  for  the  use  of 
which  interest  is  paid. 

558.  The  Rate  of  Interest  is  the  per  cent.,  or 
number  of  hundredths,  of  the  principal,  paid  for  its  use 
for  one  year. 


30 


PEECEKTAGE. 


559.  The  Amount  is  the  sum  of  the  principal  and 
the  interest. 

560.  Legal  Interest  is  the  interest  according  to 
the  rate  per  cent,  fixed  iy  law. 

561.  Usury  is  a  higher  rate  of  interest  than  is  al- 
lowed by  law. 

563.  The  legal  rates  of  interest  in  the  different  States 
are  as  follows : 


Name  of  State. 


Alabama 

Arkansas''^ 

Arizona 

California* 

Canada  and  Ireland 
Connecticut  ... 

Colorado* 

Dakota 

Delaware 

Dist.  Columbia.  . 
England  and  France 

Florida* 

Georgia 

Idaho 

Illinois 

Indiana 

Iowa 

Kansas 

Kentucky 

Louisiana 

Maine* 

Maryland 

Massachusetts*.  . 
Michigan 


Kate. 


8% 

6% 

10% 

10% 

6% 
7% 

10% 
7% 
0% 
0% 
5% 
B% 
7% 

10% 
6% 
6% 
6% 
7% 
6% 
5% 
6% 
6% 
6% 
7% 


Any. 

Any.  I 
Any. 


Any. 
Any. 


10% 


Any. 

10% 


10% 

10% 
12% 
10% 
8% 
Any. 

Any. 
10% 


Name  of  State. 


Minnesota , 

Mississippi , 

Missouri 

Montana 

New  Hampshire, 

New  Jersey 

New  York 

North  Carolina. . 

Nebraska 

Nevada* 

Ohio 

Oregon 

Pennsylvania . . . 
Rhode  Island*.  . 
South  Carolina*. 

Tennessee 

Texas 

Utah* 

Vermont 

Virginia 

West  Virginia.  . 
Washington  T.* 

Wisconsin 

Wyoming 


Rate. 


7% 
.6% 
6% 

10% 
6% 
6% 
6% 
6% 

10% 

10% 
6% 

10% 
6% 
6% 
7% 
6% 
8% 

10% 
6% 
6% 
6% 

10% 
7% 

12% 


12% 
10% 
10% 


8% 
15% 
Any. 

8% 
12% 


Any. 
Any. 

10% 
12% 
Any. 


12% 


Any. 
10% 


1.  When  the  rate  per  cent,  is  not  specified  in  accounts,  notes, 
mortgages,  contracts,  etc.,  the  legal  rate  is  always  understood. 

2.  Where  two  rates  are  specified,  any  rate  above  the  lower,  and 
not  exceeding  the  higher,  is  allowed,  if  stipulated  in  writing. 

3.  In  the  States  marked  thus  (*)  the  rate  per  cent,  is  unlimited  if 
agreed  upon  by  the  parties  in  writing. 


IKTEKEST.  31 

563.  In  the  operations  of  interest  there  avefive  parts, 
or  elements,  namely  : 

The  Princijjal ;  the  Rate  per  Cent,  per  Annum  (for  one 
year)  ;  the  Interest  j  the  Time  for  which  the  principal  is 
lent ;  and  the  Amount,  or  sum  of  the  Prin.  and  Int. 

564.  These  terms  correspond  respectively  to  Base, 
Rate,  Percentage,  and  Amount  in  Percentage,  excluding 
Time,  which  is  an  additional  element  in  Interest. 

OBAL     EXEItCISES, 

565.  1.  At  %%,  for  1  yr.,  what  decimal  part  of  the  prin- 
cipal equals  the  interest  ?     At  5^  ?     At  8%  ?    At  12 1^  ? 

2.  What  is  the  interest  of  120  for  1  year  at  b%  ? 
Analysis. — Since  the  interest  of  any  sum  at  5%  for  1  yr.  is  .05 

of  the  principal,  the  interest  of  $20  for  1  yr.  at  5^  is  .05  of  $20,  or  $1. 

3.  What  is  the  interest  of  $50  for  1  yr.  at  5^  ?  6^  ?  7^  ? 

4.  What  is  the  interest  of  $80  for  1  yr.  at  7^  ?  8%?  10^  ? 

5.  At  7^  for  5  yr.,  what  decimal  part  of  the  principal 

equals  the  interest  ? 

Analysis.— Since  the  interest  at  7%  for  1  yr.  is  .07  of  the  prin- 
cipal, the  interest  for  5  yr.  is  5  times  .07,  or  .35  of  the  principal. 
Or,  it  is  5  times  the  interest  for  1  year. 

6.  At  6^  for  3  yr.,  what  decimal  or  fractional  part  of 
the  principal  equals  the  interest  ?  At  1%  for  6  yr.  ?  At 
h%  for  5  yr.?    At  ^%  for  2  yr.  ?    At  10^  for  4  yr.? 

7.  Find  the  interest  of  $30  for  3  yr.  at  b%. 

Analysis.— Since  the  interest  of  any  sum  at  5%  for  1  yr.  is  .05 
of  the  principal,  for  3  yr.  it  is  .15,  and  .15  of  $30  is  $4.50.  Or,  the 
interest  for  1  yr.  is  .05  of  $30,  or  $1.50,  and  for  3  yr.  it  is  3  times  as 
much,  or  $4.50. 

8.  Find  the  int.  at  6^  of  $20  for  2  yr.     Of  $40  for  3  yr. 

9.  Find  the  int.  at  8^  of  $5  for  5  yr.     Of  $10  for  1 0  yr. 


32  PERCENTAGE. 

10.  At  S%  for  2  yr.  6  mo.,  what  decimal  part  of  the 
principal  equals  the  interest  ? 

Analysis. — Since  the  interest  of  any  sum  for  1  yr.  at  8%  is  .08  of 
the  principal,  the  interest  on  the  same  for  2  yr.  6  mo.  is  2i  times  .08, 
or  .20  of  the  principal.     Or,  it  is  2^  times  the  interest  for  1  year. 

11.  At  6%  for  3  yr.  3  mo.,  what  decimal  part  of  the 
principal  equals  the  interest?     At  9%  for  3  yr.  3  mo.  ? 

12.  Find  the  int.  of  $9  for  2  yr.  4  mo.  at  7%.     At  8%. 

13.  What  is  the  int.  of  $1000  for  2  yr.  3  mo.  at  10^? 
For  4  yr.  6  mo.  ?     For  5  yr.  3  mo.  ?     For  8  mo.  ? 

566.  PRiisrciPLE. — The  interest  is  the  product  of  three 
factors  ;  namely,  the  principal,  rate  per  annum,  and  time 
{expressed  in  years  or  parts  of  a  year). 

WRITTEN     EXERCISES. 

56*7.  To  find  the  interest  or  amount  of  any  sum, 
at  any  rate  per  cent.,  for  years  and  months. 

1.  Find  the  amount  of  $97.50,  at  7^,  for  2  yr.  6  mo. 

OPERATION.  Analysis.— Since  the  interest  of 

$97.50  any  sum  at  7%  for  1  yr.  is  .07  of 

Q,v  the  principal,  the  interest  of  $97.50 

'- —  at  7%  for  1  yr.  is  .07  of  $97.50,  or 

$6.8250  Intforlyr.  $6,825;  and  the  interest  for  2  yr. 

2|-  6  mo.  is  2 1  times  the  interest  for  1 


i7n^  T  .  ,    o      a  y^'^  ''''  $17,061,  and  $17.06H  $97.50 

17.06^5   Int.  for  2  yr.  6  mo.        ''     L^^/^^i    ,;       .  4     ^ 

=  $114,561,  *he  Amount. 
97.50         Principal. 


$1145625   Amount. 

Find  the  interest  and  the  amount, 

2.  Of  $450  for  3  yr.  9  mo.  at  6%.    For  8  mo.  at  7%. 

3.  Of  $247  for  5  yr.  3  mo.  at  6^%-    For  10  mo.  at  8^ 

4.  Of  $500  for  4  yr.  2  mo.  at  10^.     For  llmo.  at  5^. 


INTEREST.  33 

EuLE. — I.  Multiply  the  principal  hy  the  rate,  and  the 
product  is  the  interest  for  1  year. 

II.  Multiply  the  interest  for  1  year  by  the  time  in  years, 
and  the  fraction  of  a  year  j  the  product  is  the  required 
interest. 

III.  Add  the  pri7icipal  to  the  interest  for  the  amount. 

Formula. — Interest  =  Principal  x  Rate  x  Time. 

Find  the  interest, 

5.  Of  $36.40  for  1  yr.  7  mo.  at  6^.     At  7^.     At  ^%. 

6.  Of  $750.50  for  3  yr.  1  mo.  at  b%.     At  8^.     At  9^. 

7.  Of  $1346.84  for  2  yr.  4  mo.  at  6J^.    At  1^%. 

8.  Of  $138.75  for  4  yr.  3  mo.  at  10^.    At  n\%. 

9.  Find  the  amount  of  $640  for  5  yr.  6  mo.  at  1%. 

10.  Find  the  amount  of  $56.64  at  S%  for  3  yr.  3  mo. 

11.  Made  a  loan  of  $1040  for  1  yr.  9  mo.  at  ^%.  How 
much  is  due  at  the  end  of  the  time  ? 

12.  If  a  note  for  $375,  on  interest  at  %%,  dated  June  10, 
1874,  be  paid  Sept.  10,  1876,  what  amount  will  be  due  ? 

568.  To  find  the  interest  on  any  sum  of  money, 
for  any  time,  at  any  rate  per  cent. 

Obvious  EELATioiiirs  betweek  Time  akd  Iktekest. 

I.  The  interest  on  any  sum  for  1  year  at  1%  is  .01  of 
the  principal. 

It  is  therefore  equal  to  the  principal  with  the  decimal  point  re- 
moved tiDO  places  to  the  left. 

II.  The  interest  for  1  mo.  is  -^  of  the  interest  for  1  yr. 

III.  The  interest  for  3  days  is  -jV,  or  ^,  of  the  interest 
for  1  month  ;  hence  any  number  of  days  may  readily  be 
reduced  to  tenths  of  a  month  by  dividing  by  3. 


34  PERCENTAGE. 

IV.  The  interest  on  any  sum  for  1  month,  multiplied 
by  the  number  of  months  and  tenths  of  a  month  in  the 
given  time,  and  the  product  by  the  number  expressing 
the  rate,  will  be  the  required  interest. 

569. 1.  Find  the  int.  of  $361.20  for  1  yr.  3  mo.24  da.  at  7^. 

OPERATION. 

$3,612       (01  of  the  Prin.)    Int.  for  1  yr.  at  1  %  (568,  I). 
.301       Int.  for  1  mo.  at  1%  (568,  II). 
15.8       Number  of  months  and  tenths  (568,  III). 


$4.7558      Int.  for  1  yr.  3  mo.  24  da.  at  IJ 

7 


$33.2906  Int.  for  1  yr.  3  mo.  24  da.  at  1%  (568,  IV). 

What  is  the  interest, 

2.  Of  $137.25  for  1  yr.  6  mo.  10  da.  at  6^  ?    At  4^  ? 

3.  Of  $510.50  for  3  yr.  7  mo.  15  da.  at  b%  ?     At  8^? 

4.  Of  $1297. 60  for  2  yr.  11  mo.  18  da.  at  t%  ?    At  7^^? 

EuLE. — I.  To  find  the  interest  for  1  yr.  at  \%, 
Remove  the  decimal  point  in  the  given  principal  two 
places  to  the  left. 

II.  To  find  the  interest  for  1  mo.  at  1%. 
Divide  the  interest  for  1  year  ly  12. 

III.  To  find  the  interest  for  any  time  at  1%. 
Multiply  the  interest  for  1  month  hy  the  number  of 

months  and  tenths  of  a  month  in  the  given  time, 

IV.  To  find  the  interest  at  any  rate  %, 

Multiply  the  interest  at  l%for  the  given  time  ly  the  num- 
ler  expressi^ig  the  given  rate, 

5.  Find  the  int.  of  $781.90  for  1  yr.  1  mo.  12  da,  at  7%. 

6.  Find  the  int.  of  $3000  for  11  mo.  21  da.  at  10^. 


IlfTEKEST.  35 

7.  What  is  the  ami  of  $1049  for  2  yr.  3  mo.  9  da.  at  (j^%  ? 

8.  What  is  the  amt.  of  $216.75  for  3  yr.  5  mo.  11  da.  at  S%  ? 

9.  Eequired  the  int.  of  $250  from  Jan.  1,  1873,  to 
May  10,  1875,  at  7^? 

10.  Eequired  the  amount  of  $408.60  from  Aug.  20  to 
Dec.  18,  1876,  at  10^? 

11.  What  is  the  interest  on  a  note  for  $515.62,  dated 
March  1,  1873,  and  payable  July  16,  1875,  at  7%? 

12.  A  man  sold  his  house  and  lot  for  $12500 ;  the 
terms  were,  $4000  in  cash  on  delivery,  $3500  in  9  mo., 
$2600  in  1  yr.  6  mo.,  and  the  balance  in  2  yr.  4  mo.,  with 
6%  interest.     What  was  the  whole  amount  paid  ? 

570 •  SIX  PER  CENT  METHOD. 

At  6%  per  annum,  the  interest  of  $1 

For  12  mo is  6  cents,  or  .06  of  the  principal. 

''     2  '^  or  ^  of  12  mo., 'a  cent,    ^^.01      ''  '' 

"     1  "  "i^"Vl    "     "\     "      "  .005    " 
''     6da.'^^''    1   "     "^^  "       ^^.001    "  " 

''     \  "  "  \"    6  da.    "  .000^'^ 

571.  Principles. — 1.  Tlie  interest  of  any  sum  at  6% 
is  ONE-HALF  as  many  hundredths  of  the  principal  as 
there  are  months  in  the  given  time, 

2.  The  interest  of  any  sum  at  6%  is  one-sixth  as 
many  thousandths  of  the  principal  as  there  are  days  in 
the  given  time. 

Thus,  the  interest  on  any  sum  at  6%  for  1  yr.  3  mo.,  or  15  mo., 
is  J  of  .15,  or  .075,  of  the  principal ;  and  for  18  da.  it  is  -J  of  .018, 
or  .003,  of  the  principal.  Hence,  for  1  yr.  3  mo.  18  da.,  it  is  .075 
+  .003  =  .078  of  the  principal. 

It  is  evident  that  an  odd  month  is  ^  of  .01,  or  .005;  and  that 
any  number  of  days  less  than  6  is  such  a  fractional  part  of  .001  as 
the  days  are  of  6  days. 


36  PERCENTAGE. 

oraij    exercises. 

573.  What  is  the  interest, 

1.  Of  $1  at  Q%  for  i  year  ?    2  years  ?    3  years  ?    5  years  ? 
8  years  ?     12  years  ? 

2.  Of  $1  at  6^  for  1  month  ?   2  mo.  ?    3  mo.  ?   4  mo.  ? 
5mo.  ?    7mo.  ?    9  mo.  ?    10  mo.?   15mo.?18mo.  ? 

At  6^,  what  is  the  interest, 

3.  Of  $1  for  1  yr.  4  mo.  ?     1  yr.  7  mo.  ?     2  yr.  2  mo.  ? 

4.  Of  $1  for  1  day  ?    6  da.  ?    12  da.  ?    19  da.  ?    24  da.  ? 
33  da.?     36  da.?     45  da.?     63  da.? 

5.  Of  $1  for  1  mo.  12  da.  ?    For  3  mo.  15  da.  ?    For 
6  mo.  25  da.  ?     For  7  mo.  11  da.  ?     For  11  mo.  ]8  da.  ? 

Find  the  interest, 

6.  Of  $1,  at  Q%,  for  1  yr.  3  mo.  6  da.     For  1  yr.  9  mo. 
18  da.     For  1  yr.  5  mo.  19  da. 

7.  Of  $1  at  %%  for  2  yr.  1  mo.  9  da.    For  3  yr.  24  da. 

8.  Of  $1  at  6^  for  5  yr.  5  mo.  5  da.  For  4  yr.  7  mo.  10  da. 
At  6^,  find  the  interest, 

9.  Of  $1  for  2  yr.  6  mo.    Of  $2.     Of  $3.     Of  $5. 

10.  Of  $1  for  4  yr.  2  mo.     Of  $10.     Of  $20.     Of  $30. 

11.  Of  $5  for  1  yr.  4  mo.     For  2  yr.     For  2  yr.  8  mo. 

12.  Of  $1  for  33  da.    For  63  da.    For  93  da.    For  123  da. 

13.  Of  $6  for  33  da.     Of  $4  for  63  da.     Of  $2  for  93  da. 

14.  If  the  interest  of  a  certain  principal  at  6^  is  $18, 
what  would  the  interest  be  at  b%  ?     7;^ ?    8^ ?    9^ ?. 

5%  is  I  less  than  6%  ;  7%  is  J  more  than  6%  ;  8%  is  J  more,  etc. 

15.  If  the  interest  of  a  certain  principal  is  $16,  what 
tvould  the  int.  be  at  Z% ?    ^%?    6%?    U% ?    8^ ?    12^ ? 

16.  If  the  interest  of  a  certain  principal  is  $30,  what 
would  the  int.  be  at  2^?    4=%?    7^?    8^?    10^?    14^? 


IKTEKEST.  37 


WRITTEN      EXEItCISES. 

573.  1.  What  is  the  int.  of  $427.20  at  6^  for  2  yr.  5  mo. 

27  da. 

OPERATION.  Analysis.— Since  the  in- 

2  yr.  5  mo.  =.  29  mo.       $427.20     *^^^»*  «^  ^^  ^^^  ^  y^-  ^  ^^• 

/  _      ^  ^'^  ^^-  ^^  $149|,  or  of  any 

^  of  .29        =  .145  AA:^     g^j^  jg  1491  ^f  ^^^  princi- 

1^  of  .027     =  .004^       163.8664     pal  (571),  $427.20  x  .149J- 
Int.    =.149i0fthePrin.      =163.866+ is  the  required 

interest. 


Find  the  interest  at  6^  of 

2.  $597.25  for  7  mo.  18  da. 

3.  $418.75  for  1  mo.  25  da. 

4.  $309.18  for  2  yr.  24  da. 


5.  $1298  for  3  yr.  1  mo.  13  da. 

6.  $2000  for  2  yr.  7  mo.  24  da. 

7.  $4010  for  1  yr.  1  mo.  13  da. 


EuLE. — MuUijjly  the  given  principal  by  the  decimal  ex- 
pressing the  i?iterest  of  $1 ;  or  by  the  decimal  expressing 
one-half  as  many  hundredths  as  there  are  months,  and  one- 
sixth  as  many  thousandths  as  there  are  days,  in  the  given 
time,  and  the  product  will  be  the  required  interest. 

To  find  the  interest  at  any  other  per  cent,  by  this  method,  increase 
or  diminish  the  interest  at  6  %  by  such  part  of  itself  as  the  given 
rate  is  greater  or  less  than  6%. 

574.  To  compute  Accurate  Interest^  that  is, 
reckoning  365  da.  to  the  year,  use  the  following 

EuLE. — Find  the  interest  for  years  and  aliquot  parts  of 
a  year  by  the  common  method,  and  for  days  take  such  part 
ofl  yearns  interest  as  the  number  of  days  is  o/'365.     Or, 

When  the  time  is  in  days  and  less  than  1  yea.r,find  the 
interest  by  the  common  method  and  then  subtract  -^  part 
of  itself  for  the  common  year,  or  -^^  if  it  be  a  leap  year* 


38  PEKCENTAGE. 

1.  Find  the  accurate  interest  of  $1560  for  45  da.  at  1%. 
The  exact  int.  of  $1560  for  45  da.  at  7 fc  =  $109^x_45  ^  ^^^ ^ ^ 
Or,  It  is  $13.65  -  ^'-^'t^  ""  ^  =  $13.46  +. 

2.^  Find  the  exact  int.  of  $1600  for  1  yr.  3  mo.  at  6%, 

3.  What  is  the  difference  between  the  exact  interest  of 
$648.40  at  S%  for  1  yr.  3  mo.  20  da.  and  the  interest 
reckoned  by  the  6^  method? 

4.  Find  the  exact  interest  of  $875.60  at  7%  for  63  da. 

5.  Eequired  the  exact  interest  on  three  U.  S.  Bonds  of 
$1000  each,  at  6%,  from  May  1  to  Oct.  15. 

6.  What  is  the  exact  interest  on  a  $500  U.  S.  Bond,  at 
5%,  from  Nov.  1  to  April  10  following  ? 

575.  Find  the  interest,  by  any  of  the  ordinary  methods, 

1.  Of  $721.56  for  1  yr.  4  mo.  10  da.  at  6^. 

2.  Of  $54.75  for  3  yr.  24  da.  at  5%. 

3.  Of  $1000  for  11  mo.  18  da.  at  7^. 

4.  Of  $3046  for  7  mo.  26  da.  at  S%. 

5.  Of  $1821.50  from  April  1  to  Nov.  12  at  6%. 

6.  Of  $700  from  Jan.  15  to  Aug.  1  at  10^. 

7.  Of  $316.84  from  Oct.  20  to  March  10  at  11%. 

What  is  the  amount 

8.  Of  $3146  for  2  yr.  3  mo.  10  da.  at  1%? 

9.  Of  $96.85  for  3  yr.  1  mo.  27  da.  at  6%  ? 

10.  Of  $1008.S0  for  10  mo.  16  da.  at  Qi%  ? 

11.  Of  $2000  for  15  da.  at  12^^  ? 

12.  Of  $137.60  for  127  da.  at  10^? 

^13.  If  $1671.64  be  placed  at  interest  June  1,  1874,  what 
amount  will  be  due  April  1,  1876,  at  11%  ? 


INTEREST.  33 

14.  How  much  is  the  interest  on  a  note  for  $600,  dated 
Feb.  1,  1872,  and  payable  Sept.  25,  1875,  at  6^  ? 

15.  If  a  man  borrow  $9700  in  New  York,  and  loan  it 
in  Colorado,  what  will  it  gain  at  legal  int.  in  a  year  r 

16.  Eequired  the  interest  of  $127.36  from  Dec.  12^  1873, 
to  July  3,  1875,  at  4^^. 

17.  A  note  of  $250,  dated  June  5,  1874,  was  paid  Feb. 
14,  1875,  with  interest  at  S%.     What  was  the  amount  ^^ 

18.  A  note  for  $710.50,  with  interest  after  3  mo.,  at  7^, 
was  given  Jan.  1,  1874,  and  paid  Aug.  12,  1876.  What 
was  the  amount  due  ? 

19.  A  man  engaged  in  business  was  making  12^%  an- 
nually on  his  capital  of  $16840.  He  quit  his  business 
and  loaned  his  money  at  7^%.  What  did  he  lose  in  2  yr. 
3  mo.  18  da.  by  the  change  ? 

20.  A  man  borrows  $2876.75,  which  belongs  to  a  minor 
who  is  16  yr.  5  mo.  10  da.  old,  and  he  is  to  retain  it  until 
the  owner  is  21  years  old.  What  will  then  be  due  at  8% 
simple  interest  ? 

21.  A  speculator  borrowed  $9675,  at  6%,  April  15, 1874, 
with  which  he  purchased  flour  at  $6.35.  May  10,  1875, 
he  sold  the  flour  at  $7f  a  barrel,  cash.  What  did  he  gain 
by  the  transaction  ? 

■^  22.  A  man  borrows  $10000  in  Boston  at  6^,  reckoning 
360  da.  to  the  year,  and  lends  it  in  Ohio  at  8^,  reckoning 
365  da.  to  the  year.  What  will  be  his  gain  in  146  days? 
23.  A  tract  of  land  containing  450  acres  was  bought  at 
$36  an  acre,  the  money  paid  for  it  being  loaned  at  6^%. 
At  the  end  of  3  jr.  8  mo.  24  da.,  f  of  the  land  was  sold 
at  $40  an  acre,  and  the  remainder  at  $38|^  an  acre.  What 
was  gained  or  lost  by  the  transaction  ? 


4D  PERCENTAGE. 

^^  PKOBLEMS  IN  INTEEEST. 

576«  Interest,  time,  and  rate  given,  to  find  the 
principal. 

OBAZ    EXJERCISES, 

1.  What  sum  of  money  will  gain  $10  in  1  yr.  at  b%  ? 

Analysis. — The  interest  of  $1  for  1  yr.  at  5%  is  .05  of  the  prin- 
dpal,  and  therefore  $10  -r-  .05,  or  $200,  is  the  required  sum.    Or, 

Since  $.05  is  the  interest  of  $1,  $10  is  the  interest  of  as  many- 
dollars  as  $.05  is  contained  times  in  $10,  or  200  times.    Hence,  etc. 

What  sum  of  money  will  gain. 


2.  $20  int.  in  2  yr.  at  b%  ? 

3.  $25  int.  in  5  yr.  at  b%  ? 

4.  $60  int.  in  2  yr.  at  6%? 


5.  $84  int.  in  2  yr.  at  1%  ? 

6.  $50  int.  in  6  mo.  at  10^? 

7.  $30int.  inSmo.  at  8;^? 


WRITTEN    EXERCISES, 

577.  1.  What  sum  of  money,  put  at  interest  3|  yr.  at 
&%,  will  gain  $346.50? 

OPERATION. 

Int.  of  $1  for  3|  yr.  at  6%  =  $.21.        Analysis.— Same  as  in 
$346.50  -^  $.21  =  1650  times  ;  ^^^^  exercises.    (576.) 

$1  X  1650  =:  $1650. 

What  principal 

2.  Will  gain  $49.50  in  1  yr.  3  mo.  at  6%  ?    At  5^  ? 

3.  Will  gain  $153.75  in  3  mo.  24  da.  at  7^  ?    At  8^? 

EuLE. — Divide  the  given  interest  ly  the  interest  of  $1 
for  the  given  time,  at  the  given  rate. 
Formula. — Principal  =  Interest  -r-  {Bate  x  Time). 

What  sum  of  money 
^  4.  Will  gain  $213  in  5  yr.  10  mo.  20  da.  at  7%  ? 
5.  Willgain$173.97in4yr.4mo.  at6^?    At  12^? 


^ 


IKTEREST.  41 

6.*  A  man  receives  semi-annually  $350  int.  on  a  mort- 
gage at  1%,    What  is  the  amount  of  the  mortgage  ? 

578.  Amount,  rate,  and  time  given,  to  find  the 
principal. 

OltATj    EXEHCIS  ES. 

1.  What  sum  of  money  will  amount  to  $107  in  1  yr. 

at  7^? 

Analysis. — Since  the  interest  is  .07  of  the  principal,  the  amount 
is  1.07,  or  igj,  of  it.  If  $107  is  igj  of  the  principal,  yi^  of  the  prin- 
cipal is  y^7  of  $107,  or  $1 ;  and  ^gg,  or  the  principal  itself,  is  $100.  Or, 

Since  $1.07  is  the  amount  of  $1,  $107  is  the  amount  of  as  many 
dollars  as  $1.07  is  contained  times  in  $107,  or  $100. 

What  sum  of  money  will  amount  to 


2.  $130  in  5  yr.  at  6^? 

3.  $228  in  2  yr.  at  1%  ? 

4.  $412  in  6  mo.  at  6^? 


5.  $250  in  10  yr.  at  10^? 

6.  $350  in  15  yr.  at  5^? 

7.  $260  in  3  yr.  9  mo.  at  8^? 


WJtITTEN    EXEBCISES  , 

579.  1.  What  sum  will  amount  to  $337.50  in  5  yr. 

at  7^? 

OPERATION. 

Am't  of  $1  for  5  yr.  at  7%  =  $1.35.       Analysis.  —  Same  as 
$337.50  -^  $1.35  :r3  250  times  ;  in  oral  exercises.  (578.) 

$1  X  250  =  $250. 

What  principal 

2.  Will  amount  to  $1028  in  4  mo.  24  da.  at  7%? 

3.  Will  amount  to  $1596  in  2  yr.  6  mo.  at  6^%  ? 
^4.  Will  amount  to  $1531.50  in  3  mo.  18  da.  at  7%? 

Rule. — Divide  the  given  amount  hy  the  amou7it  of  $1 
for  the  given  time,  at  the  given  rate. 

Formula. — Prin.  =  Amt.  -r-  (1  +  Bate  x  Time). 


43  PERCENTAGE. 

5.  What  is  the  principal  which  in  217  days,  at  6^%, 
amounts  to  $918.73  ? 

6.  What  principal  in  3  yr.  4  mo.  24  da.  will  amount 
to  $761.44  at  5^? 

580.  Principal,  interest,  and  time  given,  to  find 
the  rate. 

ORAL   bxehcises, 

1.  At  what  rate  will  $100  gain  $14  in  2  years? 

Analysis. — Since  the  interest  of  $100  is  $14  for  2  yr.,  it  is  $7  for 
1  yr.,  and  $7  is  .07  of  $100,  the  principal.   Hence  the  rate  is  7  % .   Or, 

Since  the  interest  of  $100  for  2  yr.  at  1  %  is  $2,  $14  is  as  many 
per  cent,  as  $2  is  contained  times  in  $14,  or  1%. 


At  what  rate  will 
.2.  $300  gain  $60  in  4  yr.  ? 

3.  $500  gain  $100  in  5  yr.  r 

4.  $400  gain  $84  in  3  yr.  ? 


5.  $5  gain  $1  in  3  yr.  ? 

6.  $120  gain  $60  in  10  yr.? 

7.  $150  double  itself  in  10  yr.? 


WRITTEN    EXERCISES, 

581.  1.  At  what  rate  per  cent,  will  $1600  gain  $280 
interest  in  '^  years  ?         . 

OPERATION. 

Int.  of  $1600  at  1%  for  ^  yr.  =  $40.      ANALYsis.-Same  as 
$280  -  $40  =.  7  tiihes^;  1%  x  7=7^.    '(^^oT      '^'''''''' 
At  what  rate  per  cent 

2.  Will  $2085  gain  $68.11  in  5  mo.  18  da.  ? 

3.  Will  $1500  gain  $252  in  2  yr.  4  mo.  24  da.  ? 

EuLE. — Divide  the  given  interest  hy  the  interest  of  the 
^iven  principal,  for  the  given  time,  at  1%. 

Formula. — Rate  —  Int,  -r-  {Prin.  x  1%  x  Time). 


INTEREST.  43 

4.  A  house  that  cost  114500  rents  for  $1189.   What  per 
cent,  does  it  pay  on  the  investment  ? 
^^   5.  At   what  rate  will  $1500   amount  to   $1684.50  in 
2  yr.  18  da.  ? 

6.  At  what  rate  per  fnonth  will  $2000  gain  $120  in 
90  da.  ? 

7.  A  man  invests  $15600,  which  gives  him  an  annual 
income  of  $1620.     What  rate  of  interest  does  he  receive  ? 

8.  At  what  rate  per  annum  will  any  sum  double  itself 
in  4,  6,  8,  and  10  years,  respectively  ? 

At  1  % ,  any  sum  will  double  itself  in  100  yr.  ;  hence,  to  double 
itself  in  4  yr.,  the  rate  will  be  as  many  times  1^  as  4  yr.  are  con- 
tained times  in  100  yr  ,  or  25%,  etc. 

9.  At  what  rate  per  annum  will  any  sum  triple  itself 
in  2,  5,  7,  12,  and  20  years,  respectively  ? 

10.  I  invest  $49500  in  a  business  that  pays  me  $297  a 
month.     What  annual  rate  of  interest  do  I  receive  ? 

11.  Which  is  tiie  better  investment,  and  how  much, 
one  of  $4200,  yielding  $168  semi-annually,  or  one  of 
$7500,  producing  $712^  annually  ? 

■■'  ^ 

583.  Principal,  interest,  and  rate  g^iven,  to  find 
the  time. 

OJRAIj     JEXER  CIS  es, 

1.  In  what  time  will  $200  gain  $56  at  7^  ? 

Analysis.— The  given  interest,  $56,  is  -f^^,  or  .28,  of  the  princi- 
pal, $200;  therefore,  the  time  is  as  many  years  as  .07,  the  given 
rate,  is  contained  times  in  .28,  or  4  times.     Hence,  etc. 

Or,  the  interest  of  $200  at  7%  for  1  yr.  is  $14;  therefore,  the 
time  is  as  many  years  as  $14  are  contained  times  in  the  given  inter- 
est, $50,  or  4  years.     Hence,  etc. 


44  PERCENTAGE. 


In  what  time  will 

2.  $40  gain  $10  at  6%  ? 

3.  $500  gain  $100  at  4^? 

4.  $25  gain  $20  at  6^? 


5.  $1000  gain  $250  at  6%  ? 

6.  $5  gain  90  cents  at  6%  ? 

7.  $50  gain  $12|  at  10^  ? 


WRITTEN  EXEnc IS i:s, 

583.  L  In  what  time  will  $840  gain  $78.12  at  6^? 

OPERATION. 

$840  X  .06  z=  $50.40  Int.  for  1  yr.       Analysis.— Same  as  in  tb^ 
$78.12-T-$50.40i=1.55.  oral  exercises.    (582.) 

1  yr.  X  1.55  =  1  yr.  6  mo.  18  da. 

In  what  time 

2.  Will  $175.12  gain  $6.43  at  6^? 

3.  Will  $1000  amount  to  $1500  at  7|-^? 

KuLE. — Divide  the  given  interest  hy  the  interest  of  th$ 
given  principal,  at  the  given  rate  for  1  year. 
Formula. — Time  =z  Literest  —  {Prin.  x  Rate), 

4  In  what  time  will  $8750  gain  $1260  at  2%  a  month? 
V  5.  How  long  must  $1301,64  be  on  interest  to  amount 
to  $1522.92  at  5^? 

6.  How  long  will  it  take  any  sum  of  money  to  double 
itself  at  ^%,  6fc,  6%,  1^%,  and  10^,  respectively? 

At  100  % ,  any  sum  of  money  will  double  itself  in  1  year ;  hence 
to  double  itself  at  10%,  it  will  require  as  many  years  as  10%  is 
contained  times  in  100%,  or  10  yr. 

7.  How  long  will  it  take  any  sunju  to  triple  itself  at 
^%,  ^%y  7^,  ^%,  and  l^%,  respectively  ? 

8.  In  what  time  will  the  interest  of  $120,  at  %%,  equal 
the  priacipal  ?  Equal  half  the  principal  ?  Equal  twice 
the  principal  ? 


I2!^^TEREST.  45 

/;  COMPOUND    INTEREST. 

584.  Cofnpound  Interest  is  interest  not  only  on 
the  principal,  but  on  the  interest  added  to  the  principal 
when  it  becomes  due  ? 

ORAL      EXERCISES, 

585.  1.  What  is  the  comp.  int.  of  $500  in  2  yr.  at  6^  ? 

Analysis. — The  simple  interest  of  $500  for  2  yr.  is  $60 ;  the  in- 
terest of  the  first  year's  interest,  $30,  for  the  second  year  is  $1.80, 
which,  added  to  $60,  gives  $61.80,  the  compound  interest.    Or, 

The  interest  of  $500  for  1  yr.  at  6%  is  $30,  and  the  amount  is 
$530,  which  is  the  principal  for  the  second  year ;  the  interest  of  $530 
for  1  yr.  at  6%  is  $31.80,  which  added  to  $530  gives  $561.80,  the 
final  amount ;  and  deducting  $500,  the  original  principal,  gives 
$61.80,  the  compound  interest. 

What  is  the  compotmd  interest 
2.  Of  $600  for  2  yr.  at  b%  ?      4.  Of  $300  for  2  yr.  at  10^? 


3.  Of  $100  for  2  yr.  at  1%  ? 


5.  Of  $1000  for  2  yr.  at  5^? 


What  is  the  amount  at  compound  interest. 


6.  Of  $800  for  2  yr.  at  h%  ? 


8.  Of  $400  for  2  yr.  at  4^? 


7.  Of  $2000  for  2  yr.  at  10^?    9.  Of  $500  for  2  yr.  at  8^  ? 

WRITTEN     EXAMPLES, 

586.  1.  What  is  the  comp.  int.  of  $750  for  2  yr.  at  6^? 

OPERATION.  Analysis.— Since  the  amount  is  1.06 

$750   Prin.  for  let  yr.  of  the  principal,  the  amount  at  the  end 

1.06  ®^  *^^  fi^s*  y^^^  ^s  $795,  which  is  the 

'ZZZZ       ,  principal  for  the  2d  year,  and  the  amount 

$795    Prm.for  2dyr.  ^^  ^j^^  ^^^   ^^  j,^^  2^   ^^^^  .^   ^^3  ^(, 

ll_l^  Hence,  by  subtracting  the  given  princi- 

$842.70   Total  amount.       V^\  $750,  the  result  is  the   compound 
.j^^Q  t-  interest,  $92.70. 

$92.70    Compound  int. 


46  PEKCEKTAGE. 

2.  What  will  $350  amt.  to  in  3  yr.  at  7^,  comp.  int.  ? 

3.  What  is  the  compound  int.  of  $1200  for  3  yr.  at  6%  ? 

KuLE. — I.  Find  the  amount  of  the  given  principal  for 
the  first  period  of  time  at  the  end  of  ivhich  interest  is  due, 
and  make  it  the  principal  for  the  second  period, 

II.  Find  the  amount  of  this  principal  for  the  next  period; 
and  so  continue  till  the  end  of  the  given  time. 

III.  SuMract  the  given  principal  from  the  last  amounty 
and  the  remainder  will  be  the  compound  interest. 

When  the  time  contains  months  and  days,  less  than  a  single 
period,  find  the  amount  up  to  the  end  of  the  last  period,  and  com- 
pute the  simple  interest  upon  that  amount  for  the  remaining  months 
and  days,  which  add  to  find  the  total  amount. 

4.  What  will  $864.50  amount  to  in  4  yr.  at  8%^  com- 
pound interest  ? 

5.  What  is  the  compound  interest  of  $680  for  2  yr.  at 
7^,  interest  being  payable  semi-annually  ? 

6.  What  is  the  compound  interest  of  $460  for  1  yr. 
5  mo.  18  da.  at  6%,  interest  payable  quarterly  ? 

7.  What  will  be  the  amount  of  $1250  in  3  yr.  7  mo. 
18  da.  at  5%,  interest  being  semi-annual? 

8.  Find  the  compound  interest  of  $790  for  9  mo.  27  da. 
at  S%,  payable  quarterly. 

The  computation  of  compound  interest  may  be  abridged  by 
1  sing  the  following  table. 

To  use  the  table,  multiply  the  given  principal  by  the  number  in 
the  table  corresponding  to  the  given  number  of  years  and  the  given 
rate.  If  the  interest  is  not  annual,  reduce  the  time  to  periods,  and 
the  rate  proportionally.  Thus,  2  yr.  6  mo. ,  by  semi-annual  payments, 
at  7%,  is  the  same  as  5  yr.  at  3.^%  ;  and  1  yr.  9  mo.,  quarterly 
payments,  at  8^,  the  same  as  7  yr.  at  25^. 


INTEREST. 


47 


587.  Table  showing  the  arnt.  of  $1,  at  2^,  3,  3^,  4,  5,  6,  7, 
8,  9, 10,  11,  and  12%,  compound int,,fro7n  1  to  20  years. 


Yrs. 

2i  percent. 

3  per  cent. 

3i  per  cent. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

1 

1.025000 

1.030000 

1.035000 

1.040000 

1.050000 

1.060000 

2 

1.050625 

1.060900 

1.071225 

1.081600 

1.102500 

1.123600 

3 

1.076891 

1.092727 

1.108718 

1.124864 

1.157625 

1.191016 

4 

1.103813 

1.125509 

1.147523 

1.169859 

1.215506 

1.262477 

5 

1.131408 

1.159274 

1.187686 

1.216653 

1.276282 

1.338226 

6 

1.159693 

1.194052 

1.229255 

1.265319 

1.340096 

1.418519 

7 

1.188686 

1.229874 

1.272279 

1.315932 

1.407100 

1.503630 

8 

1.218403 

1.266770 

1.316809 

1.368569 

1.477455 

1.59S848 

9 

1.248863 

1.304773 

1.362897 

1.423312 

1.551328 

1.689479 

10 

1.280085 

1.343916 

J.410599 

1.480244 

1.628895 

1.790848 

11 

1.312087 

1.384234 

1.459970 

1.539454 

1.710339 

1.898299 

12 

1344889 

1.425761 

1.511069 

l.n01032. 

1.795856 

2.012197 

13 

1.378511 

1.468534 

1.563956 

1.66^074 

1.885649 

2.132928 

14 

1.412974 

1.512590 

1.618695 

1.731676 

1.9';9932 

2200904 

15 

1.448298 

1.557967 

1.675349 

1.800944 

2.078^.28 

2.396558 

16 

1.484506 

1.604703 

1.733986 

1.872981 

2.182875 

2.540852 

17 

1.521618 

1.652848 

1  794676 

1.947901 

2.292018 

2.692773 

18 

1  559659 

1.702433 

1.857489 

2.025817 

2.4C6619 

2.854339 

19 

1.598650 

1.753506 

1922501 

2.106849 

2.526950 

3.025600 

20 

1.638616 

1.806111 

1.989789 

2.191123 

2.653298 

3.207136 

Yrs. 

7  per  cent. 

8  per  cent. 

9  per  cent. 

10  per  cent. 

11  per  cent. 

12  per  cent. 

1 

1.070000 

1.080000 

1. Of.  0000 

I.IOCOOO 

I.IICOCO 

1.120000 

2 

1.144900 

1.166400 

1.188100 

1.210000 

1.232100 

1.254400 

3 

1.225043 

1.259712 

1.295029 

1.3310C0 

1.867631 

1.404908 

4 

1  310796 

1.360489 

1.411582 

1.464100 

1.516070 

1.573519 

5 

1.402552 

1.469328 

1.538624 

1.610510 

1.685058 

1.762342 

6 

1.500730 

1.586874 

1  677100 

1.771561 

1.870414 

1.973822 

7 

1.605781 

1.713824 

1.828039 

1.948717 

2.076160 

2.210681 

8 

1.718186 

1.850930 

1.992563 

2.143589 

2.804537 

2.47^963 

9 

1.838459 

1.999005 

2.171893 

2.357948 

2.558036 

2.773078 

10 

1.967151 

2.158925 

2.367364 

2.593742 

2.839420 

3.105848 

11 

2.104852 

2.331639 

2.580426 

2.853117 

3.151757 

3  478549 

12 

2.252192 

2.518170 

2.812665 

3.138428 

3.498450 

3  895975 

13 

2.409845 

2.719624 

3065805 

3.452271 

3.888279 

4.363492 

14 

2.578534 

2.937194 

3.341727 

3.797498 

4.310440 

4.887111 

15 

2.759031 

3.172169 

3.642482 

4.177248 

4.784588 

5.473565 

16 

2.952164 

3.425943 

3.970300 

4.594973 

5.310893 

6.130392 

17 

3.158815 

3.700018 

4.327633 

5.054470 

5.895091 

6.666040 

18 

3.379932 

3.996019 

4.717120 

5.559917 

6.54;551 

7.689964 

19 

3.616527 

4.315701 

5.141661 

6.115909 

7.263342 

8.612760 

20 

3.869684 

4.660957 

5.604411 

6.727500 

8.062309 

9.646291 

48  PERCENTAGE. 

9.  Find  by  the  table  the  compound  interest  of  $950  for 
1  yr.  5  mo.  24  da.,  at  10^,  interest  payable  quarterly. 

OPERATION. 

1  yr.  5  mo.  24  da.  =  5  quarters  of  a  year +  2  mo.  24  da. 
10%  per  annum  =  2|  %  per  quarter. 
Amount  for  5  yr.  at  2^%  =  1.131408  of  principal. 
$950  X  1.131408  =  $1074.837,  amount  for  1  yr.  3  mo. 
Interest  of  $1074.837  for  2  mo.  24  da.  at  10  fo  =  $25,079. 
$1074.837 +  $25,079  =  $1099.916,  total  amount. 
$1099.916  —  $950  =  $149,916,  compound  interest. 

10.  Find  the  amount,  at  compound  interest,  of  $749.25 
for  10  yr.  4  mo.,  at  7%,  interest  payable  semi-annually. 

11.  What  sum  placed  at  simple  interest  for  3  yr.  lOmo. 
18  da.,  at  11%,  will  amount  to  the  same  as  $1500  placed  at 
compound  interest  for  the  same  time,  and  at  the  same 
rate,  payable  semi-annually  ? 

rl2.  At  S%,  interest  compounded  quarterly,  how  much 
will  $850  amount  to  in  1  yr.  10  mo.  20  da.  ?. 

13.  What  will  $500  amount  to  in  20  yr.  at  7^,  comp.  int.? 

14.  A  father  at  his  death  left  $12500  for  the  benefit  of 
his  only  son,  14  yr.  8  mo.  12  da.  old,  the  money  to  be  paid 
him  when  he  should  be  21  years  of  age,  with  6%  interest 
compounded  semi-annually.     What  did  he  receive  ? 

ANI^UAL    INTEREST. 

588.  Annual  Interest  is  interest  on  the  principal 

and  on  each  year's  interest  remaining  unpaid,  but   so 

computed  as  not  to  increase  the  original  principal. 

It  is  allowed  in  the  case  of  promissory  notes  and  other  contracts 
which  contain  the  words,  "  with  interest  payable  annually,"  or  with 
"  compound  interest. "  In  such  cases,  the  interest  is  not  compounded 
beyond  the  second  year. 


INTEREST.  49 


WMITTEN    EXERCISES, 

589.  1.  Find  the  annual  interest  and  amount  of  $8000 
for  5  yr.,  at  6^  per  annum. 

OPERATION.  Analysis.— The  in- 

Int.  of  $8000  for  5  yr.  at  6^=:$2400.    t^rest  on  $8000  for  i 

-    -  $480  for  10  yr.  at  %%  =  $288.     f '  f  ^^^.  ^^Jf^^>  ^^^ 
•^  ^  for  5  yr.  is  $2400. 

$2400  +  $288==$2688,  Annual  int.  The  interest  for  the 

$8000 +  $2688=1  $10688,  Amount.  first  year,   remaining 

unpaid,  draws  interest 

for  4  yr.  ;  that  for  the  second  year,  for  3  yr.  ;  that  for  the  third  year, 

for  2  yr.  ;  and  that  for  the  fourth  year,  for  1  yr.,  the  sum  of  which 

is  equal  to  the  interest  of  $480  for  4  yr.  +  3  yr.  +  2  yr.  + 1  yr.  =  10  yr. ; 

and  the  interest  of  $480  at  6%  for  10  yr.  is  $288.    Hence  the  total 

amount  of  interest  is  $2400  +  $288,  or  $2688,  and  the  amt.  is  $10688. 

2.  What  is  the  annual  interest  of  $1500  for  4  yr.  at  7^? 

EuLE. —  Compute  the  interest  on  the  princij)dl  for  the 
given  time  and  rate,  to  which  add  the  interest  on  each 
yearns  interest  for  the  time  it  has  remained  unpaid. 

To  obtain  the  latter,  when  the  interest  has  remained 
impaid  for  a  number  of  years,  multiply  the  interest  for 
one  year  by  the  product  of  the  number  of  years  and  half 
that  number  diminished  by  one. 

Thus,  if  the  time  is  9  yr.,  the  interest  for  1  yr.  should  be  multi- 
plied by  9  X  (9  —  1)  -^  2,  or  9  x  4  =  86.  Since  the  interest  for 
the  first  year  draws  8  years'  interest,  that  for  the  second  year  7 
years'  interest,  etc.,  and  the  sum  of  the  series  8  +  7  +  6  +  5  +  4  +  3  +  3 
+  lis86. 

3.  What  will  $3500  amt.  to  in  10  yr.,  annual  int.,  at  8^? 

4.  What  is  the  difference  between  the  annual  interest 
and  the  compound  interest  of  $2500  for  6  yr.  at  6^? 

5.  Find  the  amt.  of  $575,  at  %%  annual  int.,  for  9|-  yr. 


50  PERCEIbf  TAGE. 

6.     $800.  Macon,  June  15,  ia72. 

Four  years  after  date,  for  value  received,  I  promise  to  pay 
Robert  E.  Park,  or  order,  eight  hundred  dollars,  with  in- 
terest  at  seven  per  cent,,  payable  annually. 

J.  W.  Burke. 

What  amount  is  due  on  this  note  at  maturity,  no  in- 
terest haying  been  paid  ? 

PARTIAL    PAYMEl!TTS. 

590.  Partial  Faj/iuents  are  payments  in  part  of 
the  amount  of  a  note,  bond,  or  other  obligation. 

591.  Indorsenients  are  the  acknowledgment  of 
such  payments,  written  on  the  back  of  the  note,  bond, 
etc.,  stating  the  time  and  amount  of  the  same. 

593.  A  JProniissof^y  Note  is  a  written  promise  to 
pay  a  certain  sum  of  money,  on  demand  or  at  a  specified 
time. 

593.  The  Maker  or  Drawer  of  the  note  is  the 
person  who  signs  it. 

594.  The  Payee  is  the  person  to  whom,  or  to 
whose  order,  the  money  is  paid. 

595.  An  Indorser  is  a  person  who,  by  signing 
his  name  on  the  back  of  the  note,  makes  himself  respon- 
sible for  its  payment. 

596.  The  Face  of  a  note  is  the  sum  of  money  made 
payable  by  the  note. 

597.  A  Negotiable  Note  is  one  made  payal^le  to 
bearer,  or  to  any  person's  order.  When  so  made,  it  can 
be  sold  or  transferred. 


PARTIAL     PAYMENTS.  51 

WRITTEN    EXERCISES. 

1.     $800.  New  York,  Jan.  1st,  1874. 

One  year  after  date,  I  proynise  to  pay  Caleb  Barlow,  or 
order,  eight  hundred  dollars,  for  value  received,  with  in- 
terest. James  Dunlap. 

Indorsed  as  follows  :  April  1,  1874,  $10  ;  July  1,  1874, 
$35  ;  Not.  1, 1874,  $100.  What  was  there  due  Jan.  1, 1875  ? 

Analysis— The  interest  of  $800  for  3  mo.,  from  Jan.  1  to  April  1, 
at  7%,  is  $14;  am%  $814.  Since  the  payment  is  less  than  the  in- 
terest, it  cannot  be  deducted  for  a  new  principal  without  com- 
pounding the  int.,  which  is  illegal ;  hence,  find  the  int.  of  $800  to 
the  time  of  the  next  payment,  3  mo  ,  which  is  $14,  and  the  amt.  to 
that  time,  $828,  from  which  deduct  the  sum  of  the  two  payments, 
or  $45,  leaving  $783,  a  new  principal.  The  int.  of  $783  for  4  mo., 
to  Nov.  1,  is  $18.27;  amt.,  $801.27;  from  which  deduct  the  third 
payment,  $100,  leaving  $701.27,  the  next  principal,  the  amt.  of 
which  for  2  mo.,  to  Jan.  1,  1874,  is  $709.45,  sum  due  at  that  time. 

Peinciple. — The  principal  must  not  he  increased  hy  the 
addition  of  interest  due  at  the  time  of  any  payment,  so  as 
to  compound  the  interest. 

Upon  this  principle  is  based  the  rule  prescribed  by  the 
Supreme  Court  of  the  United  States  : 

U.  S.  EuLE. — I.  Find  the  amount  of  the  given  princi- 
pal to  the  time  of  the  first  payment,  and  if  this  payment 
equals  or  exceeds  the  interest  then  due,  subtract  it  from  the 
amt.  obtained,  and  treat  the  remainder  as  a  new  principal. 

II.  If  the  interest  exceed  the  payment,  find  the  amou7it 
of  the  same  principal  to  a  time  lohen  the  sum  of  the  pay- 
ments equals  or  exceeds  the  interest  then  due,  and  subtract 
the  sum  of  the  payments  from  that  amount. 

III.  Proceed  in  the  same  manner  loith  the  remai7iing 
payments. 


52  PERCEK"TAGE, 


$500.  Philadelphia,  Feb.  1,  1875. 

2.  Three  months  after  date,  I  promise  to  pay  to  J.  B 
Lippincott  &  Co.,  or  order,  five  hundred  dollars,  with 
interest,  without  defalcation.     Value  received, 

James  Mo^^roe. 

Indorsed  as  follows  :  May  1,  1875,  $40  ;  Nov.  14,  1875, 
$8;  April  1,  1876,  $18;  May  1,  1876,  130.  What  was 
due  Sept.  16,  1876  ? 

OPERATION. 

Face  of  note,  or  principal $500.00 

Interest  to  May  1, 1875,  3  mo.,  at  6%        7.50 

Amount 507.50 

Payment,  to  be  subtracted 40.00 

2cl  principal 467.50 

Int.  of  $467.50  to  Nov.  14,  1875,  6  mo.  13  da.  .    .     $15.04 

Int.  of  $467.50  to  April  1,  1876,  4  mo.  17  da.    .    .       10.67  25.71 

Amount 493.21 

Sum  of  payments,  to  be  subtracted 26.00 

3d  principal      . 467.21 

Int.  to  May  1,  1876,  1  mo 2.34 

Amount 469.55 

Payment,  to  be  subtracted 30.00 

4tli  principal 439.55 

Int.  to  Sept.  16,  1876,  4  mo.  15  da.,  .    T 9.89 

Amount  due $449.44 

3.  What  was  the  amount  due  October  25,  1873,  upon  a 
note  for  $1500,  dated  New  Orleans,  April  1,  1872,  and 
on  which  the  following  payments  were  endorsed  :  June  5, 
1872,  $300  ;  Oct.  15,  1872,  $37.75 ;  May  1,  1873,  $97.25  ; 
Aug.  6,  1873,  $495? 


PARTIAL     PAYMENTS.  53 


$700.  Detroit,  Nov.  1,  1873. 

4.  On  demandy  1  promise  to  pay  Charles  Smith,  or 
order,  seven  hundred  dollars,  with  interest.  Value  re- 
ceived, Abraham  Isaacs. 

Indorsed  as  follows  :  Dec.  5, 1873,  $75  ;  Jan.  10,  1874, 
$350;  April  11,  1874,  $11.25;  May  15,^874,  $250. 
What  was  due  Sept.  1,  1874? 


$497^A'  Chicago,  March  15,  1874. 

5.  Three  months  after  date,  tve  promise  to  pay  James 
Kelly,  or  order,  four  hundred  and  ninety-seven  ^^^  dollars, 
with  interest  at  6%,     Value  received. 

Brown,  Nichols  &  Co. 

Indorsed  as  follows  :  Nov.  3,  1874,  $57.50 ;  June  15, 
1875,  $22.25  ;  Aug.  1, 1875,  $125  ;  Sept.  15, 1875,  $175. 
What  was  due  Jan.  1, 1876? 

598.  The  following  method  of  computation  is  often 
used  by  merchants  in  the  settlement  of  notes  and  of  in^ 
terest  accounts  running  a  year  or  less  ;  hence  called  the 
Mercantile  Eule: 

I.  Find  the  amount  of  the  note  or  deU  from  its  date 
to  the  time  of  settlement. 

II.  Find  the  amount  of  each  payment  frmn  its  date 
to  the  time  of  settlement. 

III.  Subtract  the  sum  of  the  amounts  of  payments  from 
the  amount  of  the  note  or  debt. 

An  accurate  application  of  this  rule  requires  that  the  time  should  be  reduced 
to  days,  and  that  the  interest  should  be  computed  by  the  rule  for  days  (574). 

For  the  Vermont  State  method  of  computation,  and  also  of  assessing  taxes, 
see  pages  227-231. 


54  PERCENTAGE, 

1.  On  a  note  for  $600  at  1%,  dated  Feb.  15,  1874,  were 
the  following  indorsements  :  March  25,  1874,  $150  ;  June 
1, 1874,  $75 ;  Oct.  10,  1874,  $100.    What  was  due  Dec.  31, 

1874  ? 

OPERATION. 

Am't  of  $600  from  Feb.    15  to  Dec.  31,  319  da.,  $636.71 

''  ''  $150  ''  Mar.  25  ''  "  281  da.,  $158.08 
"  ''  $75  "  June  1  '*  *'  213  da.,  78.06 
**     *'  $100     "     Oct.    10  *'        '*  82  da,,       101.57        837.71 

Balance  due  Dec.  31,  1874,  $299.00 

2.  A  note  for  $950,  dated  Jan.  25,  1876,  payable  in 
9  mo.,  at  7^  interest,  had  the  following  indorsements : 
March  2,  1876,  $225  ;  May  5,  1876,  $174.19  ;  June  29, 
1876,  $187.50;  Aug.  1,  1876,  $79.15.  What  was  the 
balance  due  at  the  time  of  its  maturity  ? 

3.  Payments  were  made  on  a  debt  of  $1750,  due  April  5, 
1875,  as  follows  :  May  10,  1875,  $190  ;  July  1,  1875, 
$230  ;  Aug.  5,  1875,  $645  ;  Oct.  1,  1875,  $372.  What 
was  due  Dec.  31,  1875,  interest  at  6^  ? 

DISOOUJ^fT. 

599.  Discount  is  a  certain  percent  deducted  from 
the  price-list  of  goods,  or  an  allowance  made  for  the  pay- 
ment of  a  debt  or  other  obligation  before  it  is  due. 

600.  The  Present  Worth  of  a  debt  payable  at  a 
future  time  without  interest,  is  such  a  sum  as,  being  put 
at  legal  interest,  will  amount  to  the  debt  when  it  becomes 
due. 

601.  The  True  Discount  is  the  difference  between 
the  whole  debt  and  the  present  worth. 


DISCOUKT.  55 

OJB^X     EXERCISES, 

603.  1.  What  is  the  present  worth  of  a  debt  of  $224, 
to  be  pdid  in  2  yr.,  at  6^  ? 

Analysis. — Since  in  2  yr.,  at  6^,  the  int,  is  .12  of  the  principal, 
the  amt.  is  1.12  of  it ;  therefore,  $234,  the  debt,  is  1.12,  or  ^^f  of 
the  present  worth,  and  jgg,  or  the  present  worth  itself,  is  $200. 
Or,  since  $1.12  is  the  amt,  of  $1,  $224  is  the  amt.  of  as  many  dol- 
lars as  $1.12  is  contained  times  in  $224,  or  $200.    (578.) 

What  is  the  present  worth 

2.  Of  $315,  due  in  10  mo.,  at  6^? 

3.  Of  $570,  due  in  2  yr.,  at  7^? 

4.  Of  $408,  due  in  3  mo.,  at  8^  ?  * 

5.  Of  $51,  due  in  4  mo.,  at  H  ? 

6.  Of  $440,  due  in  2  yr.,  at  b%  ? 
Find  the  true  discount  at  6^, 

7.  Of  $1019,  due  in  3  mo.  24  da. 

8.  Of  $102.20,  due  in  4  mo.  12  da. 

9.  Of  $5035,  due  in  1  mo.  12  da. 

WRITTEN    EXERCISES, 

603.  1.  What  are  the  present  worth  and  the  true  dis* 
count,  of  $362.95,  payable  in  7  mo.  12  da.,  at  6^  ? 

OPERATION. 

Amt.  of  $1,  for  7  mo.  12^da.,  at  6^  =  $1,037 
$362.95  -T-  $1,037  =  350*times. 
$1  X  350  =  $350,  Present  Worth. 
$362.95  —  $350  =  $12.95,  True  Discount. 

Analysts.— Since  the  amount  of  $1  for  7  mo.  12  da.  at  6%  is 
$1,037  (579),  $362.95  is  the  amount  of  as  many  dollars  as  $1,037 
is  contained  times  in  $362.95,  or  350  times.  Hence  the  present 
worth  is  $350  ;  and  the  true  discount  is  $362.95  —  $350,  or  $12,95. 


56  PERCENTAGE. 


^' 


2.  What  is  the  present  worth  of  a  debt  of  $287.'}'5  to  be 
id  in  3  mo.  18  da.  at  11%  ? 

3.  What  is  the  true  discount  on  a  debt  of  $2202.90  due 
in  8  mo.  12  da.  at  7%  ? 


KuLE. — I.  Divide  the  debt  iy  the  amount  of  $1  for  the 
given  rate  and  time,  and  the  quotie7it  is  the  present  worth. 

11.  Subtract  the  j^f'esent  worth  from  the  debt,  and  the 
remainder  is  the  true  discount. 

Formula. — Present  Worth  =  Debt  -r-  Amt.  of  $1. 

Hence  the  present  worth  is  the  principal  of  which  the  true  dis- 
count is  the  interest,  anij  the  whole  debt  the  amount. 

4.  Bought  a  house  and  lot  for  $19500  cash,  and  sold 
them  for  $22000,  payable  one-fourth  in  cash  and  the  re- 
mainder in  1  yr.  6  mo.  How  much  ready  money  did  I 
gain,  computing  discount  at  6%? 

5.  A  merchant  buys  goods  for  $4200  on  4  mo.  credit, 
but  is  offered  a  discount  of  3%  for  cash.  If  money  is 
worth  ^%  a  month,  what  is  the  difference  ? 

6.  Bought  a  bill  of  lumber  amounting  to  $3500,  on 
6  mo.  credit ;  2  months  afterward  paid  on  account  $1500, 
and  1  month  later,  $1000.  Find  the  present  worth  of 
the  balance,  at  the  time  of  the  second  payment,  int.  at  7%. 

7.  A  merchant  holds  two  notes,  one  for  $356.25  due 
Dec.  1,  1875,  and  the  otheii'  for  $497.50,  due  Feb.  1,  1876. 
What  would  be  due  him  in  cash  on  both  notes  Sept.  15^ 
1875,  at  6%  ? 

8.  A  bookseller  bought  $300  worth  of  books  at  a  dis- 
count of  33^^  from  list  prices,  and  sold  them  at  the  reg- 
ular retail  price,  on  6  mo.  time.  Money  being  worth  6^, 
what  per  cent,  profit  did  he  make  ? 


DISCOUKT.  57 

9.  A  speculator  bought  230  bales  of  cotton,  each  bale 
eontainiug  470  lb.,  at  llf  cents  a  pounds  on  a  credit  of 
9  mo.  He  at  once  sold  the  cotton  for  $13000  cash,  and 
paid  the  pres.  worth  of  the  debt  at  1%.   What  was  his  gain  ? 

10.  Which  is  the  more  profitable,  to  buy  flour  at  $8.75  a 
barrel  on  6  mo.  credit,  or  at  $8.60  on  2  mo.,  money  being 
worth  7^? 

11.  A  person  sold  goods  to  the  amount  of  $3750,  15^ 
payable  in  cash,  25^  in  3  mo.,  20^  in  4  mo.,  and  the  re- 
mainder in  6  mo.  What  ready  money  would  discharge 
the  whole  debt,  money  being  worth  Q%  ? 

r  BANK  DISCOUNT.  ^ 

604.  A  Bank  is  a  corporation  chartered  by  law  for 
the  safe-keeping  and  loaning  of  money,  or  the  issuing  of 
bills  for  circulation  as  money. 

605.  Bank  Bills  or  Notes  are  promissory  notes 

issued  by  banks,  and  payable  on  demand. 

A  bank  which,  issues  notes  to  circulate  as  money  is  called  a  Bank 
of  Issue ;  one  which  lends  money  by  discounting  notes,  a  Bank  of 
Discount ;  and  one  which  takes  charge  of  money  belonging  to  other 
parties,  called  depositors,  a  Savings  Bank,  or  Bank  of  Deposit, 
Some  banks  perform  two  and  others  all  of  these  duties. 

606.  Bank  Discount  is  a  deduction  made  for 
interest  in  advancing  money  upon  a  note  not  due,  or  pay- 
ment by  a  borrower,  in  advance,  of  interest  upon  money 
loaned  to  him.  It  is  equal  to  the  interest  at  the  given 
rate  for  the  given  time  (including  the  days  »of  grace)  on 
the  whole  sum  specified  to  be  paid. 

607.  Bays  of  Grace  are  the  three  days  allowed 
by  law  for  the  payment  of  a  note  after  the  expiration  of 
the  time  specified  in  the  note.  They  are  counted  in  by 
bankers  in  discounting  notes. 


58  ^  PERCENTAGE. 

608.  The  Maturity  of  a  note  is  the  expiration  of 
the  whole  time,  including  the  days  of  grace. 

609.  The  Term  of  Discount  is  the  time  from  the 
discount  of  a  note  to  its  maturity. 

610.  A  JBank  Chech  is  a  written  order  for  money 
by  a  depositor,  upon  a  bank. 

611.  The  Proceeds  or  Avails  of  a  note  is  the 
sum  received  for  it  when  discounted,  that  is,  the  face  of 
the  note  less  the  discount. 

613.  A  Protest  is  a  formal  declaration  in  writing, 
made  by  a  Notary-Public,  at  the  request  of  the  holder  of 
a  note,  to  give  legal  notice  to  the  maker  and  the  indorsers 
of  its  non-payment. 

1.  The  failure  to  protest  a  note  on  the  third  day  of  grace  releases 
the  indorsers  from  all  obligation  to  pay  it. 

2.  If  the  third  day  of  grace  or  the  maturity  of  a  note  occurs  on 
Sunday  or  a  legal  holiday,  it  must  be  paid  on  the  day  previous. 

3.  The  transaction  of  borrowing  money  at  a  bank  is  conducted 
as  follows  :  The  borrower  presents  a  note,  either  made  or  indorsed 
by  himself,  payable  at  a  specified  time,  and  receives  for  it  a  sum 
equal  to  the  face  less  the  interest  for  the  time  it  has  to  run,  in- 
cluding the  days  of  grace.  A  note  for  discount  at  a  bank  must  be 
made  payable  to  the  order  of  some  person,  by  whom  it  must  be 
indorsed.  When  the  note  bears  interest,  the  discount  is  computed 
on  its  face  plus  the  interest  for  the  time  it  has  to  run. 

613.  Bank  discount  being  simple  interest,  the  follow- 
ing  are  corresponding  terms  : 
The  Face  of  the  Note  is  the  principal. 
The  Term  of  Discount  is  the  time. 
The  Bank  Discount  is  the  interest. 
The  Proceeds  is  the  principal  less  the  interest. 


DISCOUiTT.  59 

614.  To  find  the  bank  discount  and  proceeds  of 
a  note. 

OJtA.L      MXEJiClSBS, 

1.  What  is  the  bank  discount  on  a  note  for  $2000  due 
in  2  mo.  15  da.  at  Q%,  and  the  proceeds  ? 

Analysis. — After  adding  3  da.,  tlie  time  is  2  mo.  18  da.  ;  the  in- 
terest for  which  at  6%  is  .013  of  the  principal  ;  .013  of  $2000  is  $26, 
the  hank  disco  ant,  and  $2000  —  $26  equals  $1974,  or  the  proceeds. 

What  are  the  l)a7iJc  discount  and  the  proceeds  of  a  note 

2.  Of  $80  for  5  mo.  27  da.,  at  7^? 

3.  Of  $100  for  2  mo.  21  da.,  at  6^  ? 

4.  Of  $200  for  8  mo.  9  da.,  at  1%  ? 

5.  Of  $150  for  4  mo.  21  da.,  at  b%  ? 

6.  Of  $100  for  30  da.,  at  Q>%  ? 

WniTTBN    JEX  EB  CISES. 

615.  1.  Eequired  the  bank  discount  and  proceeds  of 
a  note  for  $1250  due  in  90  days,  at  7^. 

OPERATION. 

$1250  X. 07  ^  gg  ^  ^^^  g^^  g^^^  Discount. 

(565 

$1250  —  $22.32  =  $1227.68,  Proceeds. 

Analysis. — The  interest  of  $1^50  for  93  da.,  at  7%,  reckoning 

865  da.  to  the  year,  is  $33.32,  which  is  the  bank  discount.    If  360  da. 

are  reckoned  to  the  year,  the  bank  disc't  is  $33,604.    Deducting  the 

bank  disc't  from  the  face  of  the  note,  the  remainder  is  the  proceeds. 

Rule. — I.  Compute  the  interest  on  the  face  of  the  note 
(or  if  it  hears  interest,  on  its  amount  at  maturity),  for 
three  days  more  than  the  specified  time,  and  the  result  ns 
the  hank  discount. 

II.  Suhtract  the  discount  from  the  face  of  the  note,  or  ' 
its  amount  at  maturity,  and  the  remainder  is  tJ^e  proceeds. 


60  PERCEISTTAGE. 

2.  Wkat  is  the  bank  discount,  and  what  is  the  pro- 
ceeds of  a  note  for  $597.50,  due  in  60  da.,  at  (j%  ? 

3.  What  will  be  the  proceeds  of  a  note  for  $1G15,  due 
in  90  da.  with  interest  at  1%,  discounted  at  the  Nassau 
Bank  in  New  York  ? 

4.  Sold  a  farm,  containing  173  A.  95  P.,  for  %Q^  an 
acre,  and  received  payment  as  follows  :  $2000  cash,  and 
the  balance  in  a  note  payable  in  5  mo.  18  da.  at  1%  inter- 
est, which  was  discounted  at  a  bank.  How  much  ready 
money  did  the  farm  bring  ? 

Find  the  date  of  maturity,  the  term  of  discount ,  and 
the  proceeds  of  the  following  : 

$957^.  Chicago,  July  27,  1875. 

5.  Three  months  after  date,  I  promise  to  pay  to  the 
order  of  D.  L.  Moody,  nine  hundred  fifty-seven  and  -^^ 
dollars,  for  value  received. 

*  Discounted  Aug.  10,  at  S%.       William  Thomson". 

$916^.  San  Francisco,  Feb.  5,  1874. 

6.  Two  months  after  date,  we  jointly  and  severally 
agree  to  pay  0.  H.  Thomas^,  or  order,  nine  hundred  six- 
teen and  y^  dollars  luith  interest  at  8^,  value  received. 

Discounted  at  Marine  Bank,  James  Barnes. 

Feb.  21,  at  10^.  George  Childs. 

$1315y^.  New  York,  May  1, 1875. 

J.  Ninety  days   after  date,  I  promise  to  pay  to  the 
order  of  Ivison,  Blakeman,  Taylor  &  Co.,  one  thousand 
three  hundred  fifteen  and  ^^  dollars,  for  value  received. 
Discounted  May  15,  at  7^.  William  Hewson". 

*  Banks  usually  count  the  actual  number  of  days  in  the  given  time,  and 
"ijS  days  to  the  year. 


DISCOUNl,  61 

$1250.  Boston,  June  12,  1876. 

8.  Six  months  after  date,  I  promise  to  pay  Knight, 
Adams  &  Co.,  or  order,  twelve  hundred  fifty  dollars,  with 
interest  at  6  per  cent.,  value  received. 

Discounted  at  a  broker's,  Geo.  B.  Damok. 

Nov.  15,  at  6%. 

616.  The  proceeds  and  time  of  a  note  given,  to 
find  the  face. 

OMAJj      EXEB,C  IS  ES  . 

1.  For  what  sum  must  a  note  be  drawn,  at  2  mo.  15  da., 
•  at  6^,  so  that  the  proceeds  when  discounted  may  be  $987  ? 

Analysis. — The  bank  discount  for  3  mo.  18  da.  at  6%  is  .013  of 
the  face  of  the  note,  and  the  proceeds  must  therefore  be  1  —  .013, 
or  .987  of  the  face  ;  and  if  .987  of  the  face  is  $987,  the  whole  face 
of  the  note  is  $1000. 

Required  the  face  of  a  note,  so  that  the  proceeds  maybe 

2.  $972,  for  4  mo.  21  da.  at  7^. 

3.  $194,  for  5  mo.  27  da.  at  6^. 

4.  $97.60,  for  3  mo.  15  da.  at  %%. 

5.  $980,  for  4  mo.  21  da.  at  b%. 

6.  $184,  for  9  mo.  15  da.  at  10^. 

wit  ITT  EN     EXEHCISES. 

617.  1.  What  must  be  the  face  of  a  note  at  9  mo. 
27  da.,  interest  S%,  so  that  the  proceeds  may  be  $448  ? 

OPERATION. 

The  bank  discount  of  $1  for  10  mo.  at  8%  is  $.066|. 
The  proceeds  of  $1  =  $1  -  $.066f  or  $.933^. 
Hence  $448  -7-  .933^  =  $480,  the  face  of  the  note. 

2.  What  is  the  face  of  a  note  at  30  da.,  the  proceeds  of 
which,  when  discounted  at  bank,  at  7^,  are  $1425  ? 


62  PEECENTAGE. 

KuLE. — Divide  the  given  proceeds  dy  the  j)TOceeds  of  |1 
for  the  time  and  rate  given ;  the  quotient  is  the  face  of 
the  note. 


Formula, — Face  =  Proceeds  —  (1  —  Bate  x  Time). 

3.  Find  the  face  of  a  3  mo.  note  the  proceeds  of  which, 
discounted  at  2%  a  month,  is  $675. 

4.  The  proceeds  of  a  note  are  $1915.75,  the  time  3  mo., 
and  the  rate  of  interest  7% ;  what  is  the  face  of  the  note  ? " 

5.  Bought  merchandise  for  $2250,  cash  ;  for  what  sum 
.must  I  draw  my  note  at  3  mo.,  so  as  to  obtain  that  sum 

at  the  bank,  interest  at  7^  ? 

6.  The  avails  of  a  3  months  note,  when  discounted  at 
tl^%,  were  $315.23  ;  what  was  the  face  of  the  note  ? 

7.  For  what  sum  must  a  note  dated  April  5,  for  90  da., 
be  drawn,  so  that  when  discounted  at  7^,  on  Ajjril  21, 
the  proceeds  may  be  $650  ? 

8.  For  how  much  must  I  draw  my  note  at  90  da.,  in 
order  that  when  discounted  at  a  bank,  at  7^,  its  avails 
will  pay  for  137i  J^-  <>*  cloth  at  $2|  a  yard? 

SAVINGS-BANK  ACCOUNTS. 

618.  A  Savings-Bcmk  is  designed  chiefly  to  ac- 
commodate depositors  of  small  sums  of  money. 

Interest  is  allowed  semi-annually  on  all  sums  that  have  been  on 
deposit  for  a  certain  time,  if  not  drawn  out  before  the  regular  daj 
of  paying  interest — generally  on  the  1st  of  January  and  of  July. 

Savings-banks  generally  allow  interest  only  from  the  commence- 
ment of  each  quarter  ;  but  in  some  banks  money  deposited  pre- 
vious to  the  1st  day  of  any  month  draws  interest  from  that  date  to 
the  day  of  declaring  interest  dividends,  provided  it  has  not  been 
previously  withdrawn. 


DISCOUNT. 


63 


WniTTEN     JEXEB  CIS  ES. 

619.  1.  A  person  had  on  deposit  Jan.  1,  1874,  $150. 
His  subsequent  deposits  were,  Feb.  3,  $35  ;  March  29, 
$20  ;  April  10,  $43  ;  May  15,  $26.  His  drafts  during  the 
same  time  were,  Jan.  15,  $50 ;  Feb.  27,  $15 ;  April  19, 
$45.    What  interest  was  due  July  1st,  at  Q%  ? 

OPERATION. 


Date  of 

Balance 

SmaUestBal. 

Interest 

SmaUestBal. 

Interest  for 

Int.  paym'ts. 

1st  of  month. 

during  mo. 

for  1  month. 

dur'g  Q'rter. 

1  Quarter. 

Jan.    1 

$150 

Feb.    1 

100 

$100 

$.50 

Mar.    1 

120 

100 

.50 

Apr.   1 

140 

120 

.60 

$100 

$1.50 

May    1 

138 

138 

.69 

June   1 

164 

138 

.69 

July  1 

164 

164 

.82 

138 

2.07 

$3.80 


$3.57 


Balance  due,  with  int.  by  monthly  periods,  $167.80. 
**      **    "  quarterly      *'       $167.57. 

Analysis. — At  the  end  of  January,  tbe  balance  due  is  $100,  which 
having  been  on  deposit  for  the  month,  draws  interest  for  1  mo. ;  at 
the  end  of  February,  the  balance  is  $120  ;  but  the  smallest  balance 
during  the  month  is  $100  ;  hence  interest  is  allowed  only  on  that 
sum.  The  same  principle  applies  to  the  other  balances.  If  only 
quarterly  periods  of  interest  are  allowed,  the  interest  is  calculated 
at  the  end  of  each  quarter  on  the  smallest  balance  during  the  quar- 
ter, or,  in  this  case,  on  $100,  April  1,  and  $138,  July  1. 

2.  Find  the  balance,  due  July  1,  on  the  following 
account :  Deposits,  Jan.  15,  $175  ;  April  10,  $60  ;  May  31, 
$110.  Drafts,  March  5,  $75;  May  1,  $35  ;  June  10,  $50. 
Interest  at  6^,  from  the  1st  day  of  each  months 


64 


PERCENTAGE. 


3.  A  person  deposits  in  a  savings-bank  the  following 
sums :  Jan.  1,  $350 ;  Feb.  5,  $150 ;  March  15,  $75 ; 
May  10,  $30  ;  June  15,  $100.  During  the  same  time  he 
draws,  Jan.  15,  $150 ;  Feb.  10,  $200  ;  March  31,  $50 ; 
June  1,  $75.  What  interest  at  6%,  payable  from  the  1st 
of  each  month,  must  be  added  to  the  account  July  1  ? 

4.  Balance  the  following,  Jan.  1,  1875  :  Balance  due  to 
Margaret  Brown,  July  1,  1874,  $275.  Deposits  received 
as  follows  :  Aug.  1,  $125 ;  Sept.  15,  $57 ;  Oct.  10,  $350. 
Drafts  paid  :  July  15,  $100  ;  Sept.  1,  $150 ;  Nov.  15, 
$6S  ;  Dec.  15,  $125.  Interest  at  6%,  from  the  1st  of  each 
quarter,  July  1  and  Oct.  1. 

KuLE. — At  the  end  of  each  term  complete  the  interest  for 
the  term  on  the  smallest  balance  on  deposit  at  any  time 
during  the  quarter  ;  and  at  the  end  of  each  period  of  six 
months  add  to  the  balance  of  principal  the  whole  amount  of 
interest  due,  and  the  sum  will  he  the  principal  at  the  com- 
mencement  of  the  next  six  inonths. 

5.  How  much  was  due  Jan.  1,  1876,  on  the  following 
account,  allowing  interest,  computed  from  the  1st  of  each 
quarter,  Jan.  1  to  July  1,  at  6%  per  annum  ? 

Br.  Greenwich  Savings  Bank,  in  acct.  with  Mary  Williams.  Or. 


1874. 

1874. 

Jan.  1 

To  Cash  . 

$136 

00 

Sept.  15 

By  Check 

$75 

00 

Mar.  17 

((      u 

25 

00 

1875. 

Aug.  1 

ii                St 

87 

50 

Jan.  20 

t(        (t 

37 

50 

1875. 

Mar.  3 

it        (I 

50 

00 

June  11 

(t           a 

150 

00 

Nov.  17 

c(           a 

72 

00 

REVIEW, 


65 


630. 


SYNOPSIS  FOR  KEVIEW. 


10.  Interest. 


1.  Defs, 


j  1.  Interest.  2.  Principal.  3.  Rate. 
•    (  4.  A 


6%  Method 


■U: 


11.  COMPOUN 

Interest, 

12.  Annual 

Interest.  )  2, 


S'D  j   1, 

iT.    (    2, 

r.  (  2, 


13.  Partial 
Payments. 


14.  Discount. 


i:: 


15.  Bank  Dis- 
count. 


16.  Savings-Bank 


Amt.    5.  Legal  Int.   6.  Usury ^ 
Corresponding  Elements. 
1.  Principle.     2.  Rule,  I,  II,  III. 
Relations  between  Time  )  j   jj   jjj   j^ 
and  Interest.  )  '       '       ' 

569.    Rule,  I,  II,  III,  IV. 

Principles,  1,  2. 
Rule. 
Accurate  Interest.     Rule. 

r  576.  1.  Bule.  2.  Formula. 
Problems    J   ^78.  1.  i?i/^^.  2.  Formula. 
1  580.  1.  Rule.  2.  Formula. 
t  582.  1.  i??/^6.  2.  Formula. 
Definitions — Compound  Interest. 
Rule,  I,  II,  III. 
Definitions. 
Rule,  I,  II. 

1.  Part.  Pay'ts.  2.  Indorsem'ts. 
3.  Promissory  Note.  4.  Maker 
or  Drawer.  5.  Payee.  6.  In- 
dorser.     7.    Face  of  a  Note. 

8.  Negotiable  Note. 
U.  S.  Rule,  I.  TI.    Merc.  Rule. 

Discount.     2.  Present  Worth. 
True  Discount. 
Rule,  I,  II. 

1.  Bank.  2.  Bank  Bills  or  Notes. 
3.  Bank  Discount.  4.  Days  of 
Grace.  5.  Maturity  of  Note. 
6.  Term  of  Discount.  7.  Bank 
Check.    S.  Proceeds  or  J  vails i 

9.  Protest. 
Corresponding  Terms. 
614.    Rule,  I,  II. 
616.    Rule    , 

Accounts— Rule. 


1.  Defs. 


2.  Principle. 


Defs.  j  o* 


Defs.  < 


/y 


631.  A  Corporation  is  an  association  of  indi- 
viduals authorized  by  law  to  transact  business  as  a  single 
person. 

633.  A  Charter  is  the  legal  act  of  incorporation 
defining  the  powers  and  obligations  of  the  body  incor- 
porated. 

633.  The  Capital  Stock  of  a  corporation  is  the 
capital  or  money  contributed,  or  subscribed  to  carry  on 
the  business  of  the  company. 

634.  Certificates  of  Stock  or  Scrip  are   the 

papers  or  documents  issued  by  a  corporation,  specifying 
the  number  of  shares  of  the  joint  capital  which  the 
holders  own. 

635.  A  Share  is  one  of  the  equal  parts  into  which 
capital  stock  is  diyided. 

The  value  of  a  share  in  the  original  contribution  of  capital  varies 
in  different  companies.  In  bank,  insurance,  and  railroad  comj)^- 
nies,  it  is  usually  $100. 

636.  Stocks  is  a  general  term  applied  to  shares  of 
stock  of  various  kinds,  Government  and  State  bonds,  etc. 

Stockholders  are  the  owners  of  stock,  either  by  original  title  or  by 
subsequent  purchase.     The  stockholders  constitute  the  company. 

637.  The  Far  Value  of  stock  is  the  sum  for  which 
the  scrip  or  certificate  was  issued. 

638.  The  Market  Value  of  stock  is  the  sum  for 

which  it  can  be  sold.  « 


STOCKS.  67 

Stock  is  at  par  when  it  can  be  sold  for  its  original  or  face  value, 
or  100^  ;  it  is  above  par,  or  at  a  premium,  when  it  will  bring  more 
than  its  face  value  ;  and  it  is  below  par,  or  at  a  discount,  when  it 
sells  for  less  than  its  face  value.  Thus,  when  stock  is  at  par,  it  is 
quoted  at  100  ;  when  it  is  5%  above  par,  at  105  ;  and  when  His  5% 
below  par,  at  95. 

639.  JPremium^  Discount^  and  JBrokerage 

are  each  2i percentage  computed  upon  the  par  value  of  the 
stock  as  the  base. 

630.  A  Stock  Broker  is  a  person  who  buys  and 
sells  stocks,  either  for  himself,  or  as  the  agent  of  another. 

631.  Stock-jobbing  is  the  buying  and  selling  of 
stocks  with  the  view  to  realize  gain  from  their  rise  and 
fall  in  the  market. 

633.  An  Installment  is  a  portion  of  the  capital 
stock  required  of  the  stockholders  as  a  payment  on  their 
subscription. 

633.  An  Assessment  is  a  sum  required  of  stock- 
holders, to  meet  the  losses,  or  to  pay  the  business  expenses 
of  the  company. 

634.  A  Dividend  is  a  sum  paid  to  the  stockholders 

from  the  profits  of  the  business. 

Dividends  and  assessments  are  a  percentage  computed  upon  the 
par  value  of  the  stock  as  the  base. 

635.  Net  Earnings  are  the  moneys  left  from  the 
profits  of  a  business  after  paying  expenses,  losses,  and  the 
interest  upon  the  bonds. 

636.  A  JBond  is  a  written  instrument  securing  the 

payment  of  a  sum  of  money  at  or  before  a  specified  time. 

The  principal  bonds  dealt  in  by  brokers  are  Government,  State, 
City,  and  Railroad  bonds. 


ffS  PERCEKTAGE. 

637.  Z7.  S.  Sands  are  of  two  kinds  ;  viz.,  these 
which  are  payable  at  a  fixed  date,  and  those  which,  while 
payable  at  a  fixed  date,  may  be  paid  at  an  earlier  specified 
time,  as  the  Government  may  elect. 

1.  The  former  are  quoted  in  commercial  transactions  by  the  rate 
of  interest  which  they  bear  ;  thus,  United  States  bonds  bearing  6% 
interest  are  quoted  U,  S.  6's.  The  latter  are  quoted  in  commercial 
transactions  by  a  combination  of  the  two  dates  ;  thus,  U.  8.  5-^0's, 
or  U.  8.  6*8  5-20 y  means  bonds  of  U.  S.  bearing  6  %  interest,  and  pay- 
able at  any  time  from  5  to  20  years,  as  the  Government  may  choose. 

2.  When  it  is  necessary  to  distinguish  different  issues  bearing  the 
game  rate  of  interest,  the  year  at  which  they  become  due  is  also 
mentioned  ;  thus,  U.  8.  S's  of  '71;  JJ.  8.  S's  of  '74;  U.  8.  6'8,  5-20, 
of '84;   U.  8.  6' 8,  5-20,  of '85, 

3.  The  5-20's  were  issued  in  1862,  'G4,  '65,  '67,  and  70.  They 
bear  interest  at  6  % ,  paid  semi-annually  in  gold,  except  the  issue  of 
1870,  called  5's  of  '81-,  which  bear  int.  at  5%,  paid  quarterly  in  gold. 

4.  Bonds  issued  by  States,  cities,  etc.,  are  quoted  in  a  similar 
manner.  Thus,  8.  G.  6'8  are  bonds  bearing  6%  interest,  issued  by 
the  State  of  South  Carolina. 

638.  A  Coupon  is  a  certificate  of  interest  attached 
to  a  bond,  to  be  cut  off  and  presented  for  payment  when 
the  interest  is  due. 

639.  Currency  is  a  term  used  to  denote  the  circu- 
lating medium  employed  as  a  substitute  for  gold  and 
silver.  It  consists,  at  present,  in  the  United  States,  of 
U.  S.  Legal-tender  Notes,  or  ^^  Greenbacks,"  and  the 
Bills  issued  by  the  Nat.  Banks,  and  secured  by  U.  S.  Bonds. 

If  from  any  cause  the  paper  medium  depreciates  in  value,  gold 
becomes  an  object  of  investment,  the  same  as  stocks.  Gold  being 
of  fixed  standard  value,  its  fluctuations  in  price  indicate  changes 
in  the  value  of  the  currency.  Hence,  when  gold  is  said  to  be  at 
a  premium,  currency  is  virtually  below  par,  or  at  a  discount. 


STOCKS.  d9 

ORAZ     JSXJUBCISJSS. 

640.  1.  Find  the  cost  of  100  shares  of  Chicago  and 

Kock  Island  Eaih'oad  stock  at  90  ;  brokerage  ^%. 

Analysis. — Since  the  cost  of  one  share  is  90%  of  $100,  or  $90, 
the  cost  of  100  shares  is  100  times  $90,  or  $9000,  to  which  add  the 
brokerage,  i%  of  $10000,  or  $12|,  and  the  sum  $9012^,  is  the  entire 
cost  of  the  stock. 

2.  What  cost  50  shares  of  N.  Y.  Central  R.  R.  Stock, 
at  par  ;  brokerage,  ^%  ? 

,     3.  Find  the  cost  of  10  shares  of  Bank  Stock  at  104 ; 
brokerage  ^%\ 

4.  What  is  the  cost  of  $2000  U.  S.  6's  5-20,  at  112 ; 
brokerage  ^%  ? 

641,  1.  A  broker  has  $5010  to  invest  in  bank  stock  at 
25^  premium  ;  how  many  shares  can  he  buy,  charging  ^% 
for  brokerage  ? 

Analysis. — Since  the  stock  sells  at  25%  premium,  each  share 
with  brokerage  will  cost  $125|  ;  hence  he  can  buy  as  many  shares 
as  $125J  are  contained  times  in  $5010,  or  40  shares. 

2.  A  speculator  invested  $52000  in  Ohio  and  Missis- 
sippi E.  R.  stock  at  25|,  allowing  ^%  brokerage ;  how 
many  shares  did  he  buy  ? 

3.  If  I  invest  $2350  in  U.  S.  6's,  '81,  at  n7|,  broker- 
age ^%,  how  many  $1000  bonds  do  I  receive  ? 

643.  1.  A  man  bought  a  number  of  shares  of  mining 
stock  at  60,  and  sold  the  same  at  68,  and  gained  $800 
by  the  transaction.     How  many  shares  did  he  buy  ? 

Analysis. — Since  he  bought  at  60%  and  sold  at  68%,  he  gained 
8%  of  the  pa'r  value  ;  hence  $800  is  8%  of  $10000,  the  par  value, 
and  the  number  of  shares  at  $100  each  is  100. 


70  PERCENTAGE. 

2.  Bought  K.  R.  stock  at  90,  and  sold  at  par,  gaining 
$1000.     Required  the  number  of  shares. 

3.  I  purchased  stock  at  110  and  sold  at  98,  losing 
$1200.     How  many  shares  did  I  buy  ? 

4.  A  broker  bought  some  stock  at  par,  and  sold  it 
at  95,  losing  $2000.    How  many  shares  did  he  buy  ? 

643.  1.  What  sum  must  be  inyested  in  California  7's,^ 
at  110,  to  obtain  therefrom  an  annual  income  of  $1400  ? 

Analysis. — Since  the  annual  income  is  |7  on  each  share,  the 
number  of  shares  must  be  equal  to  $1400  -i-  $7,  or  200  shares  ;  and 
200  shares  at  $110  amount  to  $22000,  the  required  investment. 

2.  What  sum  must  I  invest  in  stock  at  115,  paying 
10^  yearly  dividends,  to  realize  an  income  of  $2000  ? 

3.  What  sum  must  be  invested  in  N.  Y.  7's  at  103|-, 
in  order  to  receive  therefrom  an  annual  income  of  $2100  ? 

644.  1.  What  per  cent,  does  money  yield  which  is 

invested  in  S%  stock  at  120  ? 

Analysis. — Since  each  share  costs  $120,  and  pays  $8  income,  the 
per  cent,  will  be  yf^,  or  ^^  of  100%,  equal  to  6f  %. 

2.  What  per  cent,  does  stock  yield  when  bought  at 
90,  paying  6%  dividends  ?   When  bought  at  75  ?  At  120  ? 

3.  What  per  cent,  of  interest  does  stock  yield,  which 
pays  6%  semi-annual  dividends,  if  bought  at  150  ?    At . 
140  ?    At  120  ? 

645.  1.  What  should  be  paid  for  stock  yielding  C)% 
dividends,  in  order  to  realize  an  annual  interest  on  the 
investment  of  8%  ? 

Analysis. — Since  the  annual  dividend  on  each  share  is  $6,  this 
must  be  8%  of  the  sum  required  ;  and  if  8%  is  $6,  1%  is  $f,  and 
100  %  is  $75.     Hence  the  stock  must  be  bought  for  75. 


STOCKS.  71 

2.  For  what  must  stock  that  pays  7^  dividends  be 
bought  to  realize  10^  interest  ?    9^  ?    S%? 

3.  For  what  should  Missouri  6's  be  bought  to  pay  6% 
interest?    6^%?    6^%?    S%? 

646.  1.  How  much  currency  can  be  bought  for  $500 

in  gold,  when  the  latter  is  at  a  premium  of  10^  ? 

Analysis.— Since  $1  in  gold  is  worth  |1,10  in  currency,  $500  in 
gold  is  worth  500  times  $1.10,  or  $550.     Hence,  etc. 

2.  How  much  currency  can  be  bought  for  $200  in 
gold,  when  the  latter  is  at  a  premium  of  d%  ? 

3.  What  is  $1000  in  gold  worth  in  currency,  when 
the  former  is  at  a  premium  of  12^  ?   Of  9^%  ?    Of  10^^  ? 

64*7.  1.  How  much  gold  can  be  bought  for  $440  in 

currency,  when  the  former  is  at  a  premium  of  10%'? 

Analysis.— Since  $1  in  gold  is  worth  $1.10  in  currency,  $440 
will  buy  as  many  dollars  in  gold  as  $1.10  is  contained  times  la 
$440,  or  $400  in  gold.     Hence,  etc 

2.  How  much  gold  selling  at  9%  premium  will  $1090 
in  currency  buy  ?    $218  ?     $654  ? 

3.  How  much  gold  at  11^  premium  will  $444  buy  ? 

'  WRITTEN     EX  EKCISJES. 

648.  Find  the  cost 

1.  Of  220  shares  of  bank  stock,  the  market  value  of 

which  is  103J,  brokerage  J^ 

Operation.— (103f%  +  J%)of  $100  =  $104,  cost  of  1  share. 
$104  X  220  =  $22880,  cost  of  220  shares.    (640.) 

Formula. — Entire  Cost  =  {Marhet  Value  of  1  Share 
+  Brokerage)  x  No.  of  Shares. 

2.  Find  the  cost  of  350  shares  of  Western  Union  Tele- 
graph stock,  market  value  97f ,  brokerage  i%. 


72  PEECENTAGE. 

3.  A  broker  bought  for  me  15  one-thousand-doUar  TJ.  S. 
5-20  bonds  at  112J,  brokerage  ^%.    What  was  their  cost  ? 

4.  My  broker  sells  for  me  125  shares  of  stock  at  127^. 
What  should  I  receive,  the  brokerage  being  J^  ? 

649.  Find  the  nuniber  of  shares 

1.  Of  bank  stock  at  105,  that  can  be  bought  for  $25260, 

including  brokerage  at  \%  ? 

Operation.— (105%  +\%)oi  $100  =  $105|^,  cost  of  1  share. 
$25260  -^  $1051  ==  240,  No.  of  shares.    (641.) 

Formula. — No.  of  Shares  =  Investment  -r-  Cost  of  1 
Share. 

2.  How  many  shares  of  N.  J.  Central  R.  E.  stock  at 
107|,  brokerage  ^%,  can  be  bought  for  $27000? 

3.  How  many  shares  of  Mo.  6's  at  97|,  brokerage  J^, 
will  $21560  purchase? 

4.  Bought  Pacific  Mail  at  29|^,  and  sold  at  31J,  paying 
\%  brokerage  each  way.    How  many  shares  will  gain  $330  ? 

Opekation.— (31J%  -29lfc)-i%  =H%,  gain. 
-^  $1.50  =  220,  No.  of  shares.     (642.) 


Formula. — No.  of  Shares  =  Wliole  Gain  or  Loss  -j- 
Gain  or  Loss  per  Share. 

5.  How  many  shares  of  stock  bought  at  97|^  and  sold 
at  102},  brokerage  \%  each  way,  will  gain  $990  ? 

6.  Lost  $1680  by  selling  N.  Y.  Central  at  101  that  cost 
104.  Brokerage  being  \%  each  way,  how  many  shares  did 
I  sell? 

7.  How  many  shares  of  the  Bank  of  Commerce  bought 
at  110|  and  sold  at  116J,  brokerage  \%  on  the  purchase 
and  the  sale,  will  gain  $1200  ? 


^  STOCKS.  73 

650.  Find  the  amount  of  investment 

1.  In  U.  S.  5's,  of  '81,  at  111,  so  as  to  realize  therefrom 
an  annual  income  of  $2500  ? 

Operation. — $2500 -^^ $5,  income  on  1  share  =  500,  No.  of  shares. 
$111,  price  of  1  share  x  500  =  $55500,  investment.  (643.) 

FoEMULA. — Investment  =  Price  of  1  Share  x  iVb.  of 
Shares, 

2.  What  sum  must  be  invested  in  Tennessee  6's  at  85, 
to  yield  an  annual  income  of  $1800  ? 

3.  How  much  money  must  be  invested  in  any  stock  at 
105|^,  which  pays  6%  semi-annual  dividends,  to  realize  an 
annual  income  of  $2000  ? 

4.  What  sum  invested  in  stock  at  $63  per  share,  will 
yield  an  income  of  $550,  the  par  value  of  each  share  being 
$50,  and  the  stock  paying  10^  annual  dividends? 

651.  Find  the  rate  per  cejit,  of  income,  realized 

1.  From  bonds  paying  8^  interest,  bought  at  110. 

Operation. — $8,  interest  per  share  -^  $110,  cost  per  share  = 
.073?3:,  or7i\%.    (644.) 

Formula. — Rate  %  of  Income  =  Interest  per  Share  -r- 
Cost  per  Share. 

2.  If  stock  paying  10^  dividends  is  at  a  premium  of 
12^^,  what  per  cent,  of  income  will  be  realized  on  an  in- 
vestment in  it  ? 

3.  Which  will  yield  the  better  income,  S%  bonds  at  110, 
or  5's  at  75  ? 

4.  Which  is  the  more  profitable,  and  how  much,  to  buy 
New  York  7's  at  105,  or  6  per  cent,  bonds  at  84? 


74  PERCEKTAGE. 

5.  What  per  cent,  of  income  does  stock  paying  ].0^ 
dividends  yield,  if  bought  at  106  ? 

6.  What  per  cent,  will  stock  which  pays  6%  dividends 
yield,  if  bought  at  a  discount  of  16%  ? 

7.  What  rate  per  cent,  of  income  shall  I  receive,  if  I 
buy  U.  S.  5's  at  a  premium  of  10^,  and  receive  payment 
at  par  in  15  years  ? 

'653.  Find  at  what  price  stock  must  he  bought 

1.  That  pays  6%  dividends,  so  as  to  realize  an  income 
of  7^%  on  the  investment. 

Operation.— .06  -f-  .075  =  .80  or  80%,  price  of  stock.    (045.) 

Formula. — Pince  of  Stock  =  Dividend  -^  Rate  of  hi- 
come, 

2.  What  must  be  paid  for  h%  bonds,  that  the  invest- 
ment may  yield  8^? 

•  3.  How  much  premium  maybe  paid  on  stock  that  pays 
10%  dividends,  so  as  to  realize  7\%  on  the  investment  ? 

,  4.  What  must  I  pay  for  Government  5's  of  '81,  that  my 
investment  may  yield  1%  ? 

*  5.  At  what  price  must  stock,  of  the  par  value  of  $50  a 
share,  and  that  pays  6%  dividends,  be  bought,  to  yield  an 
income  of  7^%  ? 

*    6.  At  what  price  must  6%  stock  be  bought,  to  pay  as 
'  good  an  income  as  8%  stock  bought  at  par  ?    As  9%  stock  ? 

653.  Find  the  value  in  currency, 
1.  Of  $3750  in  gold,  quoted  at  llOf 

Opekation.— $1.10|  X  3750= $4143. 75,  value  in  currency.  (646.) 
Formula. — Total  Value  in  Currency  =  Value  of  $1  in 
Currency  x  No.  of  Dollars  in  Gold, 


STOCKS.  75 

2.  Find  the  value  of  $4975  in  gold,  at  a  premium  of 

"  3.  What  is  the  semi-annual  interest  of  18000  6%  gold- 
bearing  bonds  worth  in  currency,  when  gold  is  at  lllf  ? 
•  4.  A  merchant  bought  a  bill  of  goods,  for  which  he 
was  to  pay  l>7000  in  currency,  or  $6625  in  gold.  Gold 
being  at  109|,  which  is  the  better  proposition,  and  how 
much  in  currency  ? 

654.  Find  the  value  in  gold, 

1.  Of  12150  in  currency,  when  gold  is  at  a  premium  of 

Operation.— $2150  h-  1.105  =  $1945.70,  value  in  gold.    (G47.) 
Formula. — Total  VaUie  in  Gold  =  AmL  of  Currency 
-r-  (1  +  Premium), 

2.  What  is  $4500  in  currency  worth  in  gold,  when  the 
latter  is  at  a  premium  of  12|;^  ?    At  \\\%  ?    At  9^^  ? 

3.  How  much  money  must  be  invested  in  U.  S.  6's  at 
111,  when  gold  is  quoted  at  llOf,  in  order  to  obtain  a 
semi-annual  income  of  $2210  in  currency? 

4.  The  Mechanics  Bank  of  New  York  having  $109737.50 
to  distribute  to  the  stockholders,  declares  a  dividend  of 
^\%  ;  what  is  the  amount  of  its  capital  ? 

5.  A  man  owns  a  house  which  rents  for  $1450,  and  the 
tax  on  which  is  2|^  on  a  valuation  of  $8500.  He  sella 
for  $15300,  and  invests  in  stock  at  90,  that  pays  7;^  divi- 
dends. Is  his  yearly  income  increased  or  diminished, 
and  how  much  ? 

6.  If  I  have  $36500  to  invest,  and  can  buy  N.  Y.  Cen- 
tral 6's  at  85,  or  K  Y.  Central  7's  at  95,  how  much  more 
profitable  will  the  latter  be  than  the  former  ? 


y 


7fi  PEKCENTAGE. 

7.  Which  is  the  better  investment,  a  mortgage  for  3  yr. 
of  $5000,  paying  1%  interest,  and  purchased  at  a  discount 
of  b%,  or  50  shares  of  stock  at  95,  paying  S%  dividends, 
and  sold  at  the  expiration  of  3  years  at  98  ? 

8.  Henry  Ivison,  through  his  broker,  invested  a  certain 
sum  of  money  in  New  York  State  6's  at  107|^,  and  twice 
as  much  in  U.  S.  5's,  of  ^81,  at  98J,  brokerage  in  each 
case  ^%.  The  annual  income  from  both  investments  was 
$3348.     How  much  did  he  invest  in  each  kind  of  stock  ? 

9.  A  gentleman  invested  $12480  current  funds  in 
U.  S.  5-20's  of  ^85,  at  104.  What  will  be  his  annual 
income  in  currency  when  gold  is  1 10  ? 

I]:^SUBAI^CE. 

655.  Insurance  is  a  contract  of  indemnity  against 
loss  or  damage.  It  is  of  two  kinds  :  insurance  on  prop- 
erty, and  insurance  on  life. 

656.  The  Insurer  or  Underwriter  is  the  party 
who  takes  the  risk  or  makes  the  contract. 

657.  The  Policy  is  the  written  contract  between  the 
parties. 

658.  The  Premium  is  the  sum  paid  for  insurance, 
and  is  a  certain  per  cent,  of  the  sum  insured. 

659.  Insurance  business  is  generally  conducted  by  Companies, 
wliich  are  either  Joint-stock  Companies,  or  Mutual  Companies, 

A  StocJc  Insurance  Company  is  one  in  which  the  capi- 
tal is  owned  by  individuals  called  stockholders.  They  alone  share 
the  profits,  and  are  liable  for  the  losses. 

A  3£utnal  Insurance  Cornj)anif  is  one  in  which  the  profits 
and  losses  are  divided  among  those  who  are  insured. 

Some  companies  are  conducted  upon  the  Stock  and  Mutual  plans 
combined  and  are  called  Mixed  Companies. 


Insurance  on  property  is  principally  of  two  kinds:  Fire 
Insurance,  and  Marine  and  Inland  Insurance. 

660.  Fire  Insurance  is  indemnity  for  loss  of 
property  by  fire. 

661.  Marine  and  Inland  Insurance  is  in- 
demnity for  loss  of  vessel  or  cargo,  by  casualties  of  navi- 
gation on  the  ocean,  or  on  inland  waters. 

Transit  Insurance  refers  to  risks  of  transportation  by  land  only, 
or  partly  by  land  and  partly  by  water.  The  same  policy  may  cover 
both  Marine  and  Transit  Insurance. 

Stock  Insurance  is  indemnity  for  the  loss  of  cattle,  horses,  etc. 
Most  insurance  companies  will  not  take  risks  to  exceed  two-thirds 
or  three-fourths  the  appraised  value  of  the  property  insured. 

When  only  a  part  of  the  property  insured  is  destroyed  or  dam- 
aged, the  insurers  are  required  to  pay  only  the  estimated  loss  ;  and 
sometimes  the  claim  is  adjusted  by  repairing  or  replacing  the 
property,  instead  of  paying  the  amount  claimed. 

662.  The  operations  are  based  on  the  principles  of 
Percentage,  the  corresponding  terms  being  as  follows  : 

1.  The  Base  is  the  amount  of  insurance. 

2.  The  Mate  is  the  per  cent,  of  premium. 

3.  The  Percentage  is  the  premium. 

ORAL      EXEMC  IS  ES. 

663.  1.  How  much  must  be  paid  for  insuring  a  house 
and  furniture  for  $4000,  at  1\%  premium  ? 

Analysis. — Since  the  premium  is  1}%,  or  ^^,  equal  to  /^  of 
the  sum  insured,  the  premium  on  $4000  will  be  ^^^f  ^^  $4000,  or 
$50.     Hence,  etc.     (510.) 

2.  What  will  be  the  annual  premium  of  insurance,  at 
i%,  on  a  building  valued  at  $8000  ? 


78  PERCENTAGE. 

3.  What  will  be  the  cost  of  insuring  a  quantity  of  flour, 
valued  at  $1500,  at  ^%  ? 

4.  What  must  be  paid  for  insuring  a  case  of  merchan- 
dise, worth  $640,  at  2^%  ? 

X  5.  A  man  owns  f  of  a  boat-load  of  corn  valued  at 
$1800,  and  insures  his  interest  at '  If  ^.  What  premium 
does  he  pay  ? 

6.  Paid  $6  for  insuring  $300  ;  what  was  the  rate  ? 

Analysis. — Since  the  premium  on  $300  is  $6,  the  premium  on 
$1  is  ^  of  $6,  or  $.02,  equal  to  2% .    Hence,  etc.    (513.) 

7.  Paid  $12  for  an  insurance  of  $800  ;  find  the  rate. 
^**v8.  Paid  $24  for  an  insurance  of  $1000  ;  find  the  rate. 

9.  At  2%,  what  amount  of  insurance  can  be  obtained 
for  $30  premium  ? 

Analysis. — Since  2%  is  yf  ^  or  ^^  of  the  amount  insured,  $30,  the 
given  premium,  is  ^^  of  the  amount  insured ;  and  $30  is  -^j^  of  50 
times  $30,  or  $1500.     Hence,  etc.    (516.) 

What  amount  of  insurance  can  be  obtained, 

10.  On  a  house,  for  $75,  at  3%  premium  ? 

11.  On  a  boat  load  of  flour,  for  $150,  at  l%r 

12.  On  a  car  load  of  horses,  for  $90,  at  ^%  ? 

13.  On  a  store  and  its  contents,  for  $105,  at  1^%  ? 

W MITTBN    EXJEJiCISJES, 

664.  Find  the  Premium 

1.  For  insuring  a  building  for  $14500,  at  1^%. 
Operation.— $14500  x  .015  =  $217,50.    (512.)   ^       ,,^ 
Formula. — Premium  =  Amount  Insured  x  Rate. 

Find  the  premium  for  insuring 

2.  A  house  valued  at  $5700,  at  f^. 

3.  Merchandise  for  $2750,  at  1%. 


INSUBAKCE.  79 

4.  A  fishing  craft,  for  $15000,  at  1^%. 

5.  If  I  take  a  risk  of  $25000,  at  If  ^,  and  re-insure  ^ 
of  it  at  21%,  what  is  my  balance  of  the  premium  ? 

665.  Find  the  Rate  of  Insurance, 

1.  If  $36  is  paid  for  an  insurance  of  $2400. 
Operation.— $36  -h  $2400  =  .015,  or  1^% .    (5 15.) 
Formula. — Rate   of  Insurance  =  Premium  -r-  Sum 

hisured. 

What  is  the  rate  of  insurance, 

2.  If  $280  is  paid  for  an  insurance  of  $16000  ? 

3.  If  $4.30  is  paid  for  an  insurance  of  $860  ? 

4.  A  tea  merchant  gets  his  vessel  insured  for  $20000 
in  the  Eoyal  Company,  at  f^,  and  for  $30000  in  the 
Globe  Company,  at  ^%,  What  rate  of  premium  does  he 
pay  on  the  whole  insurance  ? 

"  666.  To  find  the  Amount  of  Insurance. 
1.  A  speculator  paid  $262.50  for  the  insurance  of  a 

cargo  of  corn,  at  1^%.    For  what  amount  was  the  com 

insured  ? 
Operation.— $262.50  ^  .015  :=  $17500,  the  sum  insured.   (518.) 
Formula. — Su7n  Insured  =  Premium  -V  Rate. 

^^.  If  it  cost  $93.50  to  insure  a  store  for  one-half  of  its 
value,  at  \\%,  what  is  the  store  worth  ? 
^  3.  Paid  $245  insurance  at  4f  ^  on  a  shipment  of  pork, 
to  cover  |  of  its  value.    What  was  its  total  value  ? 

4.  A  merchant  shipped  a  cargo  of  flour  worth  $3597, 
from  New  York  to  Liverpool.  For  what  must  he  insure 
it  at  3 J^,  to  cover  the  value  of  the  flour  and  premium  ? 

Operation.— $3597  h-  (1  -  .03J)  or  ,9675  =  $3717.829.    (520.) 


80  PEBCEl^TAGE. 

5.  An  underwriter  agrees  to  insure  some  property  for 
enough  more  than  its  value  to  cover  the  premium,  at  the 
rate  of  26  cents  per  $100.  If  the  property  is  worth 
$22163,  what  should  be  the  amount  of  the  policy  ? 

6.  For  what  sum  must  a  policy  be  issued  to  insure  a 
dwelling-house,  valued  at  $35000,  at  ^%,  a  carriage-house 
worth  $9500,  at  |^,  and  furniture  worth  $4500,  at  |^, 
10^  being  deducted  from  the  premium,  which  is  to  be 
covered  by  the  policy  ? 

7.  A  person  insured  his  house  for  f  of  its  value  at 
40  cents  per  $100,  paying  a  premium  of  $73.50.  What 
was  the  value  of  the  house  ? 

8.  A  dealer  shipped  a  cargo  of  lumber  from  Portland 
to  New  York ;  the  amount  of  insurance,  including  the 
value  of  the  lumber  and  the  premium  paid,  at  If  ^,  was 
$25200.     What  was  the  value  of  the  lumber.?* 

9.  A  merchant  had  500  bbl.  of  flour  insured  for  80^  of 
their  cost,  at  3^%,  paying  $107.25  premium.  At  what 
price  per  barrel  must  he  sell  the  flour  to  gain 


LIFE    INSUKANCE.* 

667.  Life  Insurance  is  a  contract  by  which  a 
company  agrees  to  pay  a  certain  sum,  in  case  of  the  death 
of  the  insured  during  the  continuance  of  the  policy. 

668.  A  Term  Life  Policy  is  an  assurance  for  one 
or  more  years  specified. 

669t  A  Wlpole  Life  Policy  continues  during  the 
life  of  the  insured. 

*  See  note  at  bottom  of  page  82. 


INSURANCE.  81 

• 

Premmms  may  be  paid  annually  for  life,  or  in  5,  10,  or  more 
installments  (called  5-payment,  10-payment  policies,  etc.),  or  the 
entire  premium  may  be  paid  in  one  sum  in  advance. 

The  premium  is  computed  at  a  certain  sum  or  rate  per  $1000 
insured,  the  rate  varying  with  the  age  of  the  insured  at  the  time 
the  policy  is  issued. 

A  policy  of  endowment  is  not  in  all  respects  an  insurance  policy, 
but  is  rather  a  covenant  to  pay  a  stipulated  sum  at  the  end  of  a 
certain  period  to  the  person  named  if  living. 

Most  companies  issue  a  form  of  policy  that  combines  the  princi- 
ples of  Term  Life  Assurance  and  Simple  Endowment,  called  for 
brevity  Endowment  Policy.     Hence, 

670.  An  Undow^nent  Policy  is  one  in  which 
the  assurance  is  payable  to  the  person  insured  at  the  end 
of  a  certain  number  of  years  named,  or  to  his  heirs  if 
he  die  sooner. 

An  endowment  policy  is  really  two  policies  in  one,  and  the 
assured  pays  the  premiums  of  both. 

671.  A  Dividend  is  a  share  of  the  premiums  or 
profits  returned  to  a  policy-holder  in  a  mutual  life  in- 
surance company. 

673.  A  Table  of  Mortality  shows  how  many  per- 
sons per  1000  at  each  age  are  expected  to  die  per  annum. 

673.  A  Table  of  Hates  shows  the  premium  to  be 
charged  for  $1000  assurance  at  the  different  ages. 

Such  a  table  is  based  upon  the  table  of  mortality,  and  the  proba- 
ble rates  of  interest  for  money  invested,  with  a  margin  or  loading 
for  expenses. 

674.  The  following  condensed  table  gives  data  from 
the  American  Experience  Table  of  mortality,  and  the 
annual  premium  on  the  kinds  of  policies  most  in  use. 


82 


PERCENTAGE. 


American  Experience  Table— Mortality  and  Premiums. 


3 
^ 

ANNUAL  PREMIUM  PER  $1000. 

Life  Table. 

Endow- 

AGE. 

1 

One 

Whole  Life, 

ment 

(AND 

Year 
Term 

Term 

1 

Payments 

Payment 

Payment 

Single 
Payment. 

Life). 

10 
years. 

^ 

{Net). 

during 
life. 

for  10  yr. 
only. 

for  5  yr. 
only. 

25 

8.1 

7.75 

$19.89 

$42.56 

$73.87 

$326.58 

$103.91 

26 

8.1 

7.82 

20.40 

43.37 

75.25 

332.58 

104.03 

27 

8.2 

7.83 

29.93 

44.22 

76.69 

338.83 

104.16 

28 

8.3 

7  95 

21.48 

45.10 

78.18 

345.31 

104.29 

29 

8.3 

8.02 

22  07 

46.02 

79.74 

352.05 

104.43 

30 

8.4 

8.10 

22.70 

46.97 

81.36 

359.05 

104.58 

31 

8.5 

8.18 

23.35 

47.98 

83.05 

366.33 

104  75 

32 

8.6 

8.28 

24.05 

49.02 

84.80 

373.89 

104.92 

as 

8.7 

8.33 

24.78 

50.10 

86.62 

381.73 

105.11 

34 

8.8 

8.49 

25.56 

51.22 

88.52 

389.88 

105.31 

35 

8.9 

860 

26.38 

52.40 

90.49 

398.34 

105.53 

40 

9.8 

9.42 

31.30 

59.09 

101.58 

445.55 

106.90 

45 

11.2 

10.73 

37.97 

67.37 

115.02 

501.69 

109.07 

50 

13.8 

13.25 

47.18 

77.77 

131.21 

567.13 

112.68 

The  actual  net  cost  of  insurance  for  a  single  year  at  each  age 
given  in  the  table,  on  the  mortality  assumed,  is  as  many  dollars  and 
tenths  of  a  dollar  as  there  are  deaths,  but  discounted  for  1  year. 
Thus,  at  age  25,  deaths  8.1  per  1000,  net  cost,  which  is  $8.10,  dis- 
counted at  4|-%  by  the  insurance  law,  $7.75.  If  this  sum,  $7.75, 
is  loaded  for  expenses  at,  say  25%,  the  total  premium  for  1  year 
is  $9.69,  if  at  40%,  then  it  would  be  $10,85. 

In  a  Term  Life  Policy  the  premium  may  vary,  increasing  slightly 
each  year  of  the  term,  according  to  the  assumed  increasing  liability 
to  decease,  or  it  may  be  averaged  for  the  term  so  as  to  be  the  same 
each  year. 

Note. — As  there  is  no  uniformity  in  the  Tables  and  Methods  used 
by  different  Life  Insurance  Companies,  the  pupil  may  very  properly 
omit  this  subject 


I2S"SURANCE.  83 

WRITTEN     EXERCISES. 

675.  To  find  the  amount  of  premium 

1.  For  a  life  policy  of  $5000  issued  to  a  person  30 
years  old. 

Operation.— $22.70  x  5  =  $113.50. 

2.  For  a  life  policy  of  $7500,  age  being  45. 

'RvL^,— Multiply  the  premium  for  $1000  assurance  by 
the  number  of  thousands. 

Formula. — Premium  —  Bate  jper  $1000  x  iVb.  of  thou- 
sands, 

3.  Find  the  annual  premium  for  an  endowment  policy 
of  $10000,  payable  in  10  years,  age  35. 

4.  What  premium  must  a  man  aged  30  pay  annually 
for  life,  for  a  life  policy  of  $5000  ? 

What  premium  annually  for  10  years  ? 
What  premium  annually  for  5  years  ? 
What  premium  in  a  single  payment  ? 

OPERATION.  Analysis.— Multiply  the  rate 

$22.70x5000=    $113.50     P^^  thousand  dollars,  found  in 

»«.9rx6«oo=  .m85   L7.S';ieZ£li:S 

$81.36  X  5000  =    $406.80     pressing  the  hundreds,  tens,  and 
$359. 05  X  5000  =  $1795. 25     units  decimally. 

5.  What  annual  premium  will  a  man  aged  35  years  pay 
to  secure  an  endowment  policy  for  $5000,  payable  to  him- 
self in  10  years,  or  to  his  heirs,  if  death  occurs  before  ? 

6.  If  he  dies  at  the  beginning  of  the  ninth  year,  how  much 
will  the  assurance  cost,  reckoning  simple  interest  at  6^  ? 

7.  How  much  less  would  he  have  paid  in  the  whole  life 
(annual  payment)  plan,  interest  included  ? 


84  PERCEKTAGE. 

8.  A  man  aged  45  insures  his  life  for  $7500  on  the  sin- 
gle-payment  plan,  and  dies  3  yr.  5  mo.  afterward.  How 
much  less  would  his  insurance  have  cost  him  had  he  in- 
sured on  the  annual  payment  plan,  reckoning  int.  at  6^  ? 

9.  A  person  aged  27  takes  out  a  10-year  endowment 
policy  for  $5000  ;  the  dividends  reduce  his  annual  pre- 
miums 15^  on  the  average.  Computing  annual  interest  at 
1%  on  his  premiums,  does  he  gain  or  lose,  and  how  much  ? 

10.  A  man  aged  35  years  took  out  a  life  policy  for 
$12000,  on  the  5-payment  plan,  and  died  3  yr.  6  mo. 
afterward.  What  was  gained  to  his  estate  by  insuring, 
computing  compound  interest  on  his  payments  at  7^, 
also  adding  two  dividends  of  $95  each  ? 

P  TAXES. 

676.  A  Tax  is  a  sum  of  money  assessed  on  the  per- 
son, property,  or  income  of  an  individual,  for  any  public 
purpose. 

677.  A  Poll  Tax  or  Capitation  Tax  is  a  cer- 
tain sum  assessed  on  every  male  citizen  liable  to  taxation. 
Each  person  so  taxed  is  called  a  poll, 

678.  A  JProperty  Tax  is  a  tax  assessed  on  prop- 
erty, according  to  its  estimated,  or  assessed,  value. 

Property  is  of  two  kinds  :  Real  Property,  or  Real  Es- 
tate, and  Personal  Property. 

679.  Real  Estate  is  fixed  property  ;  such  as  houses 
and  lands. 

680.  Personal  Property  is  of  a  movable  nature  ; 
such  as  furniture,  merchandise,  ships,  cash,  notes,  mort- 
gages, stock,  etc. 


TAXES.  85 

681*  An  Assessor  is  an  officer  appointed  to  deter- 
mine the  taxable  value  of  property,  prepare  the  assess- 
ment roUs,  and  apportion  the  taxes. 

683.  A  Collector  is  an  officer  appointed  to  receive 
the  taxes. 

683.  An  Assessment  Moll  is  a  schedule,  or  list, 
containing  the  names  of  all  the  persons  liable  to  taxation 
in  the  district  or  company  to  be  assessed,  and  the  valua- 
tion of  each  person's  taxable  property. 

684.  The  Hate  of  JP^'Ojjerty  Tax  is  the  rate  per 
cent,  on  the  valuation  of  the  property  of  a  city,  town, 
or  district,  required  to  raise  a  specific  tax. 

WRITTEN    EXEMCISES. 

685.  1.  What  sum  must  be  assessed  to  raise  $836000 
net,  after  deducting  the  cost  of  collection  at  5%  ? 

Operation.— $836000  ^  .95  =  $880000.    (519.) 

FoKMULA. — Sum  to  be  raised  -—  (1  —  Bate  of  Collection) 
=  Sum  to  be  Assessed. 

2.  What  sum  must  be  assessed  to  raise  a  net  amount 
of  $11123,  and  pay  the  cost  of  collecting  at  2%  ? 

3.  In  a  certain  district,  a  school-house  is  to  be  built  at 
a  cost  of  $1 8500.  What  amount  must  be  assessed  to  cover 
this  and  the  collector's  fees  at  3%  ? 

4.  The  expense  of  building  a  public  bridge  was  $1260.52, 
which  was  defrayed  by  a  tax  upon  the  property  of  the 
town.  The  rate  of  taxation  was  3^  mills  on  a  dollar, 
and  the  collector's  commission  was  3^%.  What  was  the 
valuation  of  the  property  ? 


86 


PERCEITTAGE. 


5.  In  a  certain  town  a  tax  of  $5000  is  to  be  assessed. 
There  are  500  polls,  each  assessed  75  cents,  and  the 
valuation  of  the  taxable  property  is  $370000.  What  will 
be  the  rate  of  property  tax,  and  how  much  will  be  A's  tax, 
whose  property  is  valued  at  $7500,  and  who  pays  for  2  polls  ? 

Operation.— $.75  x  500  =  $375,  amt.  on  polls. 

$5000  -  $375  =         '*      '*  property. 
$4635  -r-  $370000  =  .0125,  rate  of  taxation. 
$7500  X  .0125  =  $93.75,  A's  property  tax. 
$93.75  +  $1.50  =  $95.25,  A's  whole  tax. 

EuLE. — I.  Find  the  amount  of  poll  tax,  if  any,  and 
subtract  it  from  the  whole  amount  to  be  assessed ;  the 
remainder  is  the  property  tax. 

II.  Divide  the  property  tax  by  the  whole  amount  of 
taxable  property  ;  the  quotient  is  the  rate  of  taxation. 

III.  Multiply  each  marCs  taxable  property  by  the  rate 
of  taxation,  and  to  the  product  add  his  poll  tax,  if  any  ; 
the  result  is  the  lohole  amount  of  his  tax. 

A  table  such  as  the  following  is  a  great  aid  in  calculating  the 
amount  of  each  person's  tax,  according  to  the  ascertained  rate. 


Assessor's  Table. 

{Rate  . 

0087.) 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$.0087 

$9 

$.0783 

$  80 

$  .696  ' 

$  700 

$  6.09 

2 

.0174 

10 

.087 

90 

.783 

800 

6.96 

3 

.0261 

20 

.174 

100 

.87 

900 

7.83 

4 

.0348 

30 

.261 

200 

1.74 

1000 

8.70 

5 

.0435 

40 

.348 

300 

2.61     i 

2000 

17.40 

6 

.0522 

50 

.435 

400 

3.48     j 

8000 

26.10 

7 

.0609 

60 

.522 

500 

4.35     ! 

4000 

34.80 

8 

.0696 

70 

.609 

600 

5.22     1 

5000 

43.50 

T  A  X  E  8 .  87 

6.  Find  by  the  table  the  tax  of  a  person  whose  property 
is  valued  at  $3475,  the  rate  being  .0087, 

Opebation.— Tax  on  $3000  =  $26.10 

*'     "       400  =      3.48 

*'     "        70  =        .609 

*'     "    5  = .0435 

"     "  $3475  =  $30.2325,  or  $30.2a 

Find  by  the  table  the  tax  of  a  person  whose  property 
^  7.  Is  $2596,  and  who  ;^ays  for  5  polls  at  $.50. 
--8.  Is  $9785,  polls  3  at  $.75. 
,-  9.  Is  $12356,  polls  4  at  $1.25. 
\L  10.  Is  $25489,  polls  5  at  $.95. 
^     11.  A  tax  of  $11384,  besides  cost  of  collection  at  S^%, 
is  to  be  raised  in  a  certain  town.     There  are  760  polls 
assessed  at   $1.25   each,  and  the  personal   property  is 
valued  at  $124000,  and  the  real  estate  at  $350000.     Find 
the  tax  rate,^make  an  assessor's  table  for  that  rate,'  and 
find  a  person's  tax,  whose  real  estate  is  valued  at  $6750, 
personal  property  at  $2500,  and  who  pays  for  3  polls. 

12.  In  the  above  town,  how  much  is  B's  tax  on  $15000 
real  estate,  $2750  personal  property,  and  5  polls  ? 

13.  What  is  C's  tax  on  $9786  and  1  poll  ? 

14.  How  much  tax  will  a  person  pay  whose  property  is 
^^assessed  at  $7500,  if  he  pays  If  ^  village  tax,  ^%  State  tax, 

and  1 J  mills  on  a  dollar  school  tax  ? 
"^-.^  15.  The  expense  of  constructing  a  bridge  was  $916.65, 
which  was  defrayed  by  a  tax  upon  the  property  of  the 
town.  The  rate  of  taxation  was  2J  mills  on  a  dollar, 
and  the  commission  for  collecting  3% ;  what  was  the 
assessed  valuation  of  the  property  of  the  town  ? 
Note. — Amt.  to  be  raised  -5-  by  rate  =  valuation. 


88 


PERCEKTAGE. 


686. 


o 


SYNOPSIS    FOE    KEVIEW. 

"  1.  Corporation.  2.  Charter.  3.  Capital  Stock, 
4.  Certificate  of  Stock,  or  Scrip.  5.  Share. 
6.  Stocks,  11.  Stockholders.  8.  P«r  Value, 
9.  Market  Value.    10.  Premium,  Discount, 

1.  Defs.  -^  Brokerage.  11.  >8^^A;  Broker.  12.  ^ocA:- 
jobbing.  13.  Installment.  14.  Assessment. 
15.  Dividend.  16.  J^ef  Earnings.  17.  Bond. 
18.  Dif.  ^m^Z*  <?/  CT.  >8^.  J?(wc?«.  19.  Cbw- 
^n.    20.  Currency. 

2.  CM8.  1  r  C6?«^. 

3.  64:9.  iV^<?.  of  Sf tares. 

4.  050.  )  ^w*.  of  Investment. 

5.  651.  >  To  find  K  i?af^  %  Income. 

6.  652.  Pwe  fo  j?ay  Income. 

7.  653.  FaZ2/e  0/  G^o^  iri  (7?/r. 

8.  654.  J  I  Fa^M6  o/Owr.  iw  (?oZd 


mula. 


H 
fe 


!1.  Insurance.     2.  Insurer  or    Underwriter, 
3.  Policy.  4.  Premium.   5.  i^*>6  Insurance. 
6.  Marine  or  Inland  Insurance. 
J   2.  Corresponding  Terms  in  Percentage. 

3.  664r.  i  (  Premium.  ^ 

4.  665.  f  To  find  ^  -Ba^6  <?/  Insurance.  V  Formula. 
^  5.  666.  )  (  ^w^.  of  Insurance.  ) 

^  1.  Life  Insurance.    2.  Term  Life  Policy.    3. 
Tfi^^6  X^/(3  Policy.    4.  Endowment  Policy. 

1.  Defs.  ^       ^,Dimdend.  6.  Table  of  Mortality.  7.  Table 

of  Bates. 

2.  675.     Rule.     Formula. 


1.  Defs. 


2.  685 
L  3.  686. 


u 


1,  r^aj.  2.  Poll  Tax.  3.  Property  Tax.  4. 
ii?6aZ  Estate.  5.  Personal  Property.  6. 
Assessor.  7.  Collector.  S.  Assessment  Boll 
9.  JSa^6  ^/  Property  Tax. 

T    fi  r1   ^  /Swm  ^(?  &6  raised.    Formula. 
I  ^m^.  ^/  Taa;.     Rule,  I,  II,  III. 


t^^ 


^^v       ^^^^       ^ 


687.  Exchange  is  the  giving  or  receiving  of  any 
sum  in  one  currency  for  its  value  in  another. 

By  means  of  exchange,  payments  are  made  to  persons  at  a  dis- 
tance by  written  orders,  called  Bills  of  Exchange. 

688.  Exchange  is  of  two  kinds.  Domestic,  or  In- 
land,  and  Foreign. 

689.  Domestic  or  Inland  Exchange  relates 
to  remittances  made  between  different  places  in  the  same 
country. 

690.  Foreign  Exchange  relates  to  remittances 
made  between  different  countries. 

691.  A  Bill  of  Exchange  is  a  written  request,  or 
order,  upon  one  person  to  pay  a  certain  sum  to  another 
person,  or  to  his  order,  at  a  specified  time.  An  inland 
bill  of  exchange  is  usually  called  a  Draft. 

693.  A  Set  of  Exchange  is  a  bill  drawn  in  dupli- 
cate or  triplicate,  each  copy  being  valid,  until  the  amount 
of  the  bill  is  paid.  These  copies  are  sent  by  different 
conveyances,  to  provide  against  mis6arriage. 

693.  A  Sight  Draft  or  Bill  is  one  which  requires 
payment  to  be  made  ^^at  sight, '^  that  is,  at  the  time  it  is 
presented  to  the  person  wjio  is  to  pay  it. 


90  •  PERCENTAGE. 

694.  A  Time  Draft  or  Bill  is  one  that  requires 
payment  to  be  made  at  a  cerlain  specified  time  after  date, 
or  after  sight. 

695.  The  Buyer  or  Meniitter^  of  a  bill  is  the 
person  who  purchases  it.  The  buyer  and  payee  may  be 
the  same  person. 

696.  The  Acceptance  of  a  bill  or  draft  is  the  agree- 
ment by  the  drawee  to  pay  it  at  maturity.  The  drawee 
thus  becomes  the  acceptor,  and  the  bill  or  draft,  an 

acceptance. 

1.  The  drawee  accepts  by  writing  the  word  "  accepted "  across 
the  face  of  the  biU,  and  signing  it. 

2.  Three  days  of  grace  are  usuaUy  aUowed  on  bills  of  exchange, 
as  well  as  on  notes.  When  a  bill  is  protested  for  non-acceptance, 
the  drawer  is  bound  to  pay  it  immediately. 

697.  The  Par  of  JExchange  is  the  estimated  value 
of  the  coins  of  one  country  as  compared  with  those  of 
another.     It  is  either  intrinsic  or  commercial, 

1.  The  Intrinsic  Par  of  Exchange  is  the  comparative  value  of  the 
coins  of  different  countries,  according  to  their  weight  and  purity. 

2.  The  Commercial  Par  of  Exchange  is  the  comparative  value  of 
the  coins  of  different  countries,  according  to  their  market  price. 

698.  The  Course  or  Bate  of  JExchange  is  the 

current  price  paid  in  one  place  for  bills  of  exchange  on 
another  place. 

This  price  varies  according  to  the  relative  conditions  of  trade  and 
commercial  credit  at  the  two  places  between  which  the  exchange  is 
made.  Thus,  if  New  York  is  largely  indebted  to  London,  bills  of 
exchange  on  London  will  bear  a  high  price  in  New  York. 


EXCSAKaE.:  "91 

699.         FORMS  OF  DRAFTS  AND  BILLS. 

A   SIGHT  DRAFT. 

$500.  New  York,  Jvly  1,  1874. 

At  sight,  pay  to  the  order  of  William  Thompson,  five 
hundred  dollars,  value  received,  and  charge  to  the  acct,  of 

He:n^ry  J.  Carpenter. 
To  Harris,  Jones  &  Co., 

Cincinnati,  0. 

Other  drafts  have  the  same  form  as.  the  aDove,  except  that  in- 
stead of  the  words  *^  at  sight,"  " days  after  sight,"  or  ** 

days  after  date/'  are  used.     When  the  time  is  after  sights  it  meana 
after  acceptance. 

SET   OF   EXCHANGE. 

i;700.  New  York,  ^w^ws«  1,1874. 

At  sight  of  this  First  of  Exchange  (Second  and  Third 
of  the  same  tenor  and  date  unpaid),  pay  to  the  order  of 
Samuel  Monmouth,  Seven  Hundred  Pounds  Sterling,  for 
value  received,  and  charge  the  same  to  the  account  of 

Morton,  Bliss  &  Co. 

Morton,  Eose  &  Co.,  London. 

The  above  is  the  form  of  ilie  first  bill ;  the  second  requires  only 
the  change  of  ''First"  into  "Second,"  and  instead  of  "Second 
and  Third  of  the  same  tenor,"  etc.,  ''  First  and  Third."  The  Third 
Bill  varies  similarly. 

DOMESTIC  OR  INLAND  EXCHANGE. 

The  course  of  exchange  for  inland  bills,  or  drafts,  is  always  ex- 
pressed  by  the  rate  of  premium  or  discount.  Time  drafts,  however, 
are  subject  to  bank  discount,  like  promissory  notes,  for  the  term 
of  credit  given.  Hence,  their  cost  is  affected  by  both  the  course  of 
exchange  and  the  rate  of  discount  for  the  time. 


92  PERCENTAGE. 

WRITTEN    EXERCISES, 

700.  What  is  the  cost 

1.  Of  a  sight  draft  on  New  Orleans  for  $1750,  at  l\% 
premium  ? 

Operation.— $1750  x  LOIJ  =  $1771.871.    (512.) 

^  ^    .        7^  S  1  +  Rcite  of  Premium, 

Formula. — Cost  =  Face  x  i  ^       t>  .     /.  r^- 

(  1  —  Rate  of  Discount. 

2.  Of  a  sight  draft  on  Troy  for  11590,  at  1^%  discount  ? 

3.  Of  a  draft  on  Boston  for  $1650,  payable  in  60  days 
after  sight,  exchange  being  at  a  premium  of  If ^  ? 

Operation. — $1.0175  =  Course  of  Exchange. 

$.0105  =  Bank  Dig.  on  $1,  for  63  da. 
$1,007    =  Cost  of  Exchange,  for  $1. 
$1,007  X  1650  =  $1661.55,  value  of  Draft. 

w-  4.  Of  a  draft  on  New  York  at  30  da.  for  $4720,  at  l^% 
premium  ? 

5.  Of  a  draft  on  New  Orleans,  at  90  da.,  for  $5275,  int. 
being  1%,  and  exchange  Y/c  discount  ? 

\v  6.  Find  the  cost  in  Philadelphia  of  a  draft  on  Denver, 
at  90  da.,  for  $6400,  the  course  of  exchange  being  lOlf? 

\^  7.  What  must  be  paid  in  New  York  for  a  draft  on 
San  Francisco,  at  90  da.,  for  $5600,  the  course  of  ex- 
change being  102^^  ? 

701.  Find  the  Face 

1.  Of  a  draft  on  St.  Louis,  at  90  da.,  purchased  for 
$4500,  exchange  being  at  101^^. 

Operation. — $1,015    =  Course  of  Exchange. 

$.0155  =  Bank  Dis.  of  $1,  for  93  da.,  at  6%. 
$.9995  =  Cost  of  Exchange  of  $1. 
$4500  -r-  .9995  =  $4502.25.    (520.) 


EXCHANGE.  93 

_^2.  Of  a  draft  on  Richmond  at  60  da.  sight,  purchased 
for  $797.50,  interest  7^,  premium  2j^%. 
-—3.  Of  a  sight  draft  bought  for  $711.90,  discount  1^%, 

4.  A  commission  merchant  sold  2780  lb.  of  cotton  at 
/11|^  cents  a  pound.     If  his  commission  is  2^%,  and  the 

course  of  exchange  9S^%,  how  large  a  draft  can  he  buy  to 
remit  to  his  consignor  ? 

5.  The  Broadway  Bank  of  New  York  having  declared 
a  dividend  of  5^,  a  stockholder  in  Chicago  drew  on  the 
bank  for  the  sum  due  him,  and  sold  the  draft  at  a  pre- 
mium of  H%y  thus  realizing  |2283.18f  from  his  dividend- 
How  many  shares  did  he  own  ? 

tJ3.  A  man  in  Rochester  purchased  a  draft  on  Louisville, 
y.,  for  $5320,  drawn  at  60  days,  paying  $5151.09.   What 
was  the  course  of  exchange  ? 

\i  7.  Received  from  Savannah  250  bales  of  cotton,  each 
weighing  520  pounds,  and  invoiced  at  12|  cents  a  pound. 
Sold  it  at  an  advance  of  25^,  commission  1^%,  and 
remitted  the  proceeds  by  draft.  What  was  the  fece  of 
the  draft,  exchange  being  ^%  discount  ? 

^  FOREIGN    EXCHANGE. 

703.  Money  of  Account  consists  of  the  denomi- 
nations or  divisions  of  money  of  any  particular  country, 
in  which  accounts  are  kept. 

The  Act  of  March  3,  1873,  provides  that  *'  the  -ralue  of  foreign 
coin,  as  expressed  in  th«  moniBj  of  account  of  the  United  States, 
shall  be  that  of  the  pure  metal  of  su«h  coin  of  standard  value  ;  and 
the  values  of  the  standard  coins  in  circulation,  of  the  various  na- 
tions of  the  world,  shaU  be  estimated  annually/  hj  the  Director  of 
the  Mint,  and  be  proclaimed  on  the  first  daj  of  Januaiy  hj  the 
Secretary  of  the  Treasury." 


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PERCEIifTAGE. 


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EXCHANGE.  95 

704.  Sterling  Bills  or  Sterling  Exchange 

are  bills  on  England,  Ireland,  or  Scotland.  Such  bills 
are  negotiated  at  a  rate  fixed  without  reference  to  the  par 
of  exchange. 

Formerly  such  bills  were  quoted  at  a  certain  rate  fc  above  the 
old  par  value  of  a  pound  sterling,  which  was  $4.44f .  As  this  was 
entirely  a  fictitious  value,  and  always  about  9%  below  the  real 
value,  the  course  of  exchange  always  appeared  to  be  heavily  against 
this  country,  and  thus  tended  to  impair  its  credit.  By  the  Act  of 
March,  1873,  **  all  contracts  made  after  the  first  day  of  January, 
1874,  based  on  an  assumed  par  of  exchange  with  Great  Britain  of 
fifty-four  pence  to  the  dollar,  or  $4.44f  to  the  sovereign  or  pound 
sterling,"  are  declared  nuU  and  void.  The  par  of  exchange  between 
Great  Britain  and  the  United  States  is  fixed  at  $4.8665. 

705.  Exchanges  with  Europe  are  effected 
chiefly  through  the  following  prominent  financial  circles: 
London,  Paris,  Antwerp,  Amsterdam,  Hamburg,  Frank- 
fort, Bremen,  and  Berlin. 

In  exchange  on  Paris,  Antwerp,  and  Switzerland,  the  unit  is 
the  franc,  and  the  quotation  shows  the  number  of  francs  and 
centimes  to  the  dollar.  Federal  Money.  In  exchange  on  Amster- 
dam, the  unit  is  the  guilder,  quoted  at  its  value  in  cents  ;  on  Ham- 
burg, Frankfort,  Bremen,  and  Berlin,  the  quotation  shows  the  value 
of  fottr  reichsmarks  (marks)  in  cents, 

WJRITTEN     EXAMJPJ^ES. 

'706.  Find  the  «>5/ 

1.  Of  a  bill  of  exchange  on  London  at  3  days'  sight, 
for  £393  15s,  6d,  exchange  being  quoted  at  4.89^,  and 
gold  at  1.10^. 

OPERATION. 

£393  15s.  ^di,  =  £393.775. 

$4,895  X  393  775  =  $1927.529,  gold  value  of  bOL 

$1927.529  X  1.10|  =  $2122.69,  value  in  mirrene^. 


rBKCCI^T  ACB. 


S.Ofaia 


EXCHANaB,  97 

13,  What  will  it  cost  to  remit  directly  from  Boston  to 
Amsterdam,  12560  guilders,  at  41  J? 

14.  What  will  be  the  cost  of  remitting  13550  marks 
from  New  York  to  Frankfort,  exchange  selling  at  94^, 
and  gold  at  lOOJ  ;  brokerage,  ^%  ? 

707.  What  will  be  the /ace 

1.  Of  a  bill  of  exchange  on  London  that  can  be  bought 

for  $5500,  in  currency,  exchange  selling  at  4.86,  and  gold 

at  1.10? 

Operation.— $5500  currency  -*- 1.10  =  $5000,  gold.    (519.) 
$5000  -J-  $-^.80  =  1028.806  -h . 
£1028.806  =  £1028  168,  li<L 

2.  Of  a  bill  on  Manche^ter^  England,  that  can  be 
bought  for  $7500,  gold  ;  rate  of  exchange,  4.86  ? 

3.  Of  a  bill  on  Berlin  that  cost  $4000  in  gold,  ex- 
change 93|^. 

Operation.— <$1000  -^  $.9375)  x  4  =  17066|  marks. 
Analysis. — Since  $.93}  will  buy  4  marks,  $4000  \^ill  buy  4  times 
as  many  marks  as  $.93}  is  contained  times  in  $4000,  or  ITOOOf  marks. 

4.  Of  a  bill  on  Hamburg  that  cost  $550  in  gold,  ex- 
change 94|  ? 

5.  Of  a  bill  on  Frankfort  that  cost  $395.75  in  gold, 
exchange  95^? 

6.  Of  a  bill  on  Geneva,  Switzerland,  that  cost  $325  in 

gold,  exchange  at  5.17? 

Operation.— 5.17  fr.  x  325  =  1680.25  francs. 
Analysis.  —If  $1  \^ill  buy  5.17  francs,  $325  will  buy  325  times 
5.17  francs,  or  1680.25  francs. 

7.  A  merchant  in  New  Orleans  gave  $6186,  currency, 
for  a  bill  on  Paris,  at  5.15^.     What  was  its  face  ? 

8.  What  is  the  face  of  a  bill  on  Antwerp,  that  may  be 
purchased  in  New  York  for  $2500,  exchange  at  5.16i? 


t)9  PERCENTAGE. 

ARBITEATION    OF    EXCHANGE. 

108.  Arbitration  of  Exchange  is  the  process  of 
computing  the  cost  of  exchange  between  two  places  by 
means  of  one  or  more  intermediate  exchanges.  Such  ex- 
change is  said  to  be  indirect,  or  circuitous. 

By  this  computation  the  relative  cost  of  direct  and  indirect  ex- 
change is  ascertained.  Sometimes,  owing  to  the  course  of  exchange 
between  different  places,  it  is  more  advantageous  to  remit  by  the 
latter  than  by  the  former. 

Arbitration  is  either  simple  or  compound. 

709.  Simple  Arbitration  is  that  in  which  there 
is  but  one  intermediate  place. 

710.  Compound  Arbitration  is  that  in  which 
there  are  several  intermediate  places. 

WniTTEN      EXERCISES. 

711.  1.  I  owe  1500  marks  to  a  merchant  in  Frankfort. 
Should  I  remit  directly  from  New  York,  or  through  Lon- 
don, exchange  on  Fra-nkfort  being  94,  on  London  4.87|^, 
and  in  the  latter  place  on  Frankfort  20. 75  marks  to  the 
pound,  and  the  London  brokerage  \%  ? 

Operation.— $.94  x  1500-^4= $352.50,  cost  of  direct  exchange. 
1500  marks  -f-  20.75  marks  =  £72.29. 

£72.29  +  4%  =£72.38. 

$4.87i  X  72.38  =  $352.85. 

$352.85  —  $352.50  =  $.35,  loss  by  ind.  exchange, 

2.  What  will  it  cost  to  remit  from  Boston  to  Berlin 
750  marks,  by  indirect  exchange,  through  Paris,  exchange 
in  New  York  on  Paris  being  at  5.15,  and  4  marks 
at  Paris  being  worth  4.91  francs,  the  brokerage  being 
ati^? 


EXCHANGE.  Cd9 

3.  What  will  it  cost  to  remit  2500  guilders  from  New 
York  to  Amsterdam,  through  London  and  Paris,  the  rates 
of  exchange  being  as  follows  :  at  New  York  on  London 
4.83,  at  London  on  Paris  24.75  francs  to  the  pound,  and 
at  Paris  on  Amsterdam  2.09  francs  to  the  guilder,  broker- 
age at  London  and  Paris  i%  each  ? 

OPERATION. 

$  X  =  2500  guilders. 

1  guilder       =  2.09  francs. 

1  franc  (net)  =  1 .00|^  (with  brokerage). 

24.75  francs  =  £1. 

£1  (net)  =  £1.00^  (with  brokerage). 

£1  =  $4.83. 

2500x2.09x1.001^x1.001x4.83 
Hence,      -, ^^ ^— ,     or 

By  cancenatio«,      100x19  xl.OOj-xl.OOjx  1.61  ^  ^^^^^^^ 

o 

Analysis. — Since  the  members  of  each  equation  are  equal,  the 
product  of  the  corresponding  members  of  any  number  of  equations 
are  equal ;  hence,  the  product  of  all  the  second  members  divided  by 
the  product  of  all  the  first  members  except  one,  must  give  that 
member,  which  is  the  value  required. 

4.  A  merchant  in  St.  Louis  directs  his  agent  in  New 
York  to  draw  upon  Philadelphia  at  1%  discount,  for 
$1500  due  from  the  sale  of  mdse.  ;  he  then  draws  upon 
the  New  York  agent,  at  2%  premium,  for  the  proceeds, 
after  allowing  the  agent  to  reserve  ^%  commission.  What 
BvCm  does  he  realize  from  his  mdse.  ? 

OPERATION. 

( a; )  St.  L.  =  1500  Philadelphia. 
100  Phil.   =      99  N.York. 
100  N.  Y.  =    102  St.  Louis. 
1  =  .995  (net  proceeds). 

By  cancellation,    .  15  x  99  x  102  x  .995 =$1507. 13. 


100  PERCEKTAGE. 

Analysis.— $100  on  Philadelphia  =  $99  on  N.  Y.,  and  $100  on 
N.  Y.  =  $102  on  St.  Louis  ;  and  since  the  agent  reserves  |%  com- 
mission, $1  realized  =  $.995  net  proceeds.  Arranging,  canceling, 
and  multiplying,  we  find  the  result  to  be  $1507.13. 

EuLE. — I.  Represent  the  required  sum  iy  {x),  tvith  the 
proper  unit  of  currency  affixed,  and  place  it  equal  to  the 
given  sum  on  the  right. 

II.  Arrange  the  given  rates  of  exchange  so  that  in  any 
two  consecutive  equations  the  same  unit  of  currency  shall 
stand  on  opposite  sides. 

III.  When  there  iscomrnissionfor  drawing,  place  1  minus 
the  rate  on  the  left  if  the  cost  of  exchange  is  required,  and 
on  the  right  if  proceeds  are  required ;  and  when  there  is 
commission  for  remitting,  place  1  plus  the  rate  on  the 
right,  if  cost  is  required,  and  on  the  left,  if  proceeds  are 
required. 

IV.  Divide  the  product  of  the  numbers  on  the  right  iy 
the  product  of  the  numbers  on  the  left,  canceling  equal  fac- 
tors, and  the  result  will  be  the  required  sum. 

Commission  for  drawing  is  commission  on  the  sale  of  a  draft ; 
commission  for  remitting  is  commission  on  the  purchase  price  of  a 
draft. 

The  above  method  of  operation  is  sometimes  called  the  Chain  Rule. 

5.  If  at  New  York  exchange  on  London  is  4.84|^,  and 
at  London  on  Paris  it  is  25.73  francs  to  the  £,  what  is 
the  arbitrated  course  of  exchange  between  New  York  and 
Paris? 

6.  If  in  London  exchange  on  Paris  is  25.71,  and  in 
New  York  on  Paris  it  is  5.15^,  what  is  the  arbitrated 
course  of  exchange  between  New  York  and  London  ? 


EXCHA:efGE.  101 

7.  A  banker  in  New  York  remits  $5000  to  Liverpool 
by  indirect  exchange,  through  Paris,  Hamburg,  and  Am- 
sterdam, the  rates  being  as  follows  :  in  New  York  on 
Paris  5.18  fr.  to  the  dollar,  in  Paris  on  Hamburg  1.22  fr. 
to  the  mark,  in  Hamburg  on  Amsterdam  1.70  mark  to  the 
guilder,  and  in  Amsterdam  11.83  guilders  to  the  pound 
sterling.  How  much  sterling  will  he  have  in  bank  at 
Liverpool,  and  how  much  does  he  gain  by  indirect  ex- 
change, sterling  being  worth  in  New  York  4.83^  ? 

8.  A  merchant  in  Philadelphia  owes  a  correspondent 
in  Paris  35000  francs.  Direct  exchange  on  Paris  is  5.15  ; 
but  exchange  on  London  is  4.83,  and  London  exchange 
on  Paris  is  25. 12.  Allowing  ^%  commission  for  brokerage 
at  London,  which  is  the  more  advantageous  way  to  remit, 
and  by  how  much  ? 

9.  An  American  resident  at  Amsterdam  wishing  to 
obtain  funds  from  the  U.  S.  to  the  amount  of  14500, 
directs  his  agent  in  London  to  draw  on  Philadelphia,  and 
remit  the  proceeds  to  him  in  a  draft  on  Amsterdam,  ex- 
change on  London  in  Phil,  selling  at  4.87^,  and  in  Lon- 
don on  Amsterdam  11.17^  guilders  to  the  pound  sterling. 
If  the  agent  charges  commission  at  ^%  both  for  drawing 
and  remitting,  how  much  better  is  this  arbitration  than 
to  draw  directly  on  the  U.  S.  at  41|^  cents  per  guilder  ? 

10.  A  speculator  residing  in  Cincinnati,  having  pur- 
chased 165  shares  of  railroad  stock  in  New  Orleans,  at 
75^,  remits  to  his  agent  in  N.  York  a  draft  purchased  at 
2%  premium,  directing  the  agent  to  remit  the  sum  due  on 
N.  Orleans.  Now,  if  exchange  on  N.  Orleans  is  at  f  ^  dis- 
count in  N.  Y.,  and  the  agent's  commission  for  remitting 
is  ^%,  how  much  does  the  stock  cost  in  Cincinnati  ? 


102  PERCENTAGE. 


^       CUSTOM-HOUSE    BUSIK^ESS. 

713.  A  Custom^House  is  an  office  established  by 
government  for  the  transaction  of  business  relating  to  the 
collection  of  customs  or  duties,  and  the  entry  and  clear- 
ance of  vessels. 

713.  A  Port  of  Entry  is  a  seaport  town  in  which 
a  custom-house  is  established. 

714.  The  Collector  of  the  JPort  is  the  officer  ap- 
pointed by  government  to  attend  to  the  collection  of 
duties  and  to  other  custom-house  business. 

715.  A  Clearance  is  a  certificate  given  by  the  Col- 
lector of  the  port,  that  a  vessel  has  been  entered  and 
cleared  according  to  law. 

By  the  entry  of  a  vessel  is  meant  the  lodgment  of  its  papers  in 
the  custom-house,  on  its  arrival  at  the  port. 

716.  A  Manifest  is  a  detailed  statement,  or  invoice, 
of  a  ship's  cargo. 

No  goods,  wares,  or  merchandise  can  be  brought  into  the  United 
States  by  any  vessel,  unless  the  master  has  on  board  a  full  mani- 
fest, showing  in  detail  the  several  items  of  the  cargo,  the  place 
where  it  was  shipped,  the  names  of  the  consignees,  etc. 

717.  Duties  or  Customs  are  taxes  levied  on  im- 
ported goods. 

The  general  object  of  such  taxes  is  the  support  of  government, 
but  they  are  also  designed  sometimes  to  protect  the  manufacturing 
industry  of  a  country  against  foreign  competition. 

718.  A  Tariff  is  a  schedule  showing  the  rates  of 
duties  fixed  by  law  on  all  kinds  of  imported  merchandise. 

Duties  are  of  two  kinds.  Specific  and  Ad  Valorem. 


CUSTOM-HOUSE     BUSIl^ESS.  103 

719.  A  Specific  Duty  is  a  fixed  sum  imposed  on 
articles  according  to  their  weight  or  measure,  but  without 
regard  to  their  value. 

*730.  An  Ad  Valorem  Duty  is  an  import  duty 

assessed  by  a  percentage  of  the  value  of  the  goods  in  the 

country  from  which  they  are  brought. 

Before  computing  specific  duties,  certain  deductions,  or  allow- ' 
ances,  are  made,  called  Tare,  Leakage,  Breakage,  etc. 

721.  Tare  is  an  allowance  for  the  weight  of  the  box,  cask, 
bag,  etc.,  that  contains  the  merchandise. 

722.  Leakage  is  an  allowance  for  waste  of  liquors  Imported 
in  casks  or  barrels. 

723.  breakage  is  an  allowance  for  loss  of  liquors  imported 
in  bottles. 

734.  Gross  Weight  or  Value  is  the  weight  or 
value  of  the  goods  before  any  allowance  is  made?. 

735.  N'et  Weight  or  Value  is  the  weight  or  value 
of  the  goods  after  all  allowances  have  been  deducted 

WniTTEN     JEXEMC  IS  ISS. 

736.  Find  the  Duty 

1.  On  355  yds.  of  carpeting,  invoiced  at  lis.  6d.  per 

yd.,  the  duty  being  50%. 

Operation.— lis.  6d.  =  £.575. 

£.575  X  355  =  £204.125. 

$4.8665  (par  value  of  £1)  x  204125  =$993.37. 

$993.37  X  JO  =  $496.68,  duty.    (510:) 

2.  On  50  hhd.  of  sugar,  each  containing  500  lb.,  at  5| 
cts.  per  lb.  ;  duty  If  ets.  per  lb. 

3.  On  350  boxes  of  cigars,  each  containing  100  cigars, 
invoiced  at  17.50  per  box,;  weight,  12  lb.  per  1000 ;  duty, 
$2.50  per  lb.,  and  25;^  ad  valorem. 


104  PEECBNTAGE. 

4.  A  wine  merchant  in  New  York  imported  from  Havre 
100  doz.  quart  bottles  of  champagne,  at  $13  per  doz.,  and 
25  casks  of  sherry  wine,  each  containing  30  gals.,  at  $2.50 
per  gal.  What  is  the  duty,  the  rate  on  the  champagne 
being  $6  per  dozen,,  and  on  the  sherry  60  cents  per  gal., 
and  25^  ad  valorem  ? 

5.  Imported  from  Geneva  25  watches  invoiced  at  $125 
each,  and  15  clocks,  at  $37.50.  What  was  the  duty,  the 
rate  being  on  clocks  25^,  and  on  watches,  35^  ad  valorem? 

6.  A  liquor  dealer  receives  an  invoice  of  120  dozen  pint 
bottles  of  porter,  rated  at  $.75  per  dozen.  If  2^%  of  the 
bottles  are  found  broken,  what  will  be  the  duty  at  36  cts. 
per  gallon  ? 

7.  H.  B.  Claflin  &  Co.  imported  20  cases  of  bleached 
muslins,  each  case  containing  175  pieces  of  24  yards 
ea<^h,  IJ  yards  wide.  What  was  the  duty  at  5^  cts.  per 
square  yard  ? 

8.  What  was  the  duty  on  10  cases  of  shawls,  average 
weight  of  eaeh  case  213|^  lb.,  invoiced  at  19375  francs ; 
rate  of  duty,  50  cts.  per  lb.  and  35^  ad  valorem  ?  If  I  pay 
fox  tha  invoice  with  a  bill  of  exchange  bought  at  5.15^, 
and  pay  charges  amounting  to  $67.50  currency,  what  do 
the  shawls  cost  me  in  currency,  gold  selling  at  1.10  ? 

9.  Olmsted  &  Taylor,  of  New  York,  import  from 
Switzerland  1  ease  of  watches,  invoiced  at  7125  francs; 
duty,  25^;  charges,  13.50  francs;  comniissions,  2^%, 
What  was  the  cost  of  the  watches  in  U.  S.  gold  ? 

10.  Imported  from  England  5  cases  of  cloths  and  cassi- 
meres,  net  weight,  695  lb. ;  value  as  per  invoice,  £375 
10s.  What  was  the  duty  in  American  gold,  the  rate 
being  50  cts.  per  lb.  and  35^  ad  valorem  ? 


EQUATIOK     OF     PAYMENTS.  105 


EQUATIOJN"    OF    PAYMENTS. 

737.  Equation  of  Payments  is  the  process  of 
finding  the  average  time  for  the  payment  of  several  sums 
of  money  due  at  different  times,  without  loss  to  debtor 
or  creditor. 

738.  The  Equated  Time  is  the  date  at  which  the 
several  debts  may  be  discharged  by  one  payment. 

739.  The  Term  of  Credit  is  the  time  at  the 
expiration  of  which  a  debt  becomes  due. 

730.  The  Avei^age  Term  of  Credit  is  the  time 
at  the  end  of  which  the  several  debts  due  at  diff'erent 
dates,  may  all  be  paid  at  once,  without  loss  to  debtor  or 
creditor. 

ORAJL      EXERCISES, 

731.  1.  The  interest  of  $100  for  3  mo.  equals  the 
interest  of  $50  for  how  many  months  ? 

Analysis. — At  the  same  rate,  the  interest  of  $100  equals  the 
interest  of  $50,  or  one-half  of  $100,  for  twice  the  time,  or  6  mo. 

2.  The  interest  of  $20  for  4  mo.  equals  the  interest  of 
$10  for  how  many  mo.  ?  Equals  the  interest  of  $5  for 
how  many  mo.  ?     Of  $1  ?     Of  $40  ?     Of  $100  ? 

3.  The  interest  of  $25  for  6  mo.  equals  the  interest  of 
$5  for  how  many  mo.  ?     Of  $10  ?     Of  $1  ? 

4.  The  interest  of  $10  for  6  mo.,  and  of  $100  for  2  mo., 
taken  together,  equals  the  interest  of  $1  for  how  many 
months?      r. 


106  PERCENTAGE, 

5.  If  I  borrow  $50  for  3  mo.,  for  how  many  months 

should  I  lend  $100  to  repay  an  equal  amount  of  interest  ? 

Analysis. — The  interest  of  $50  for  3  mo.  is  the  same  as  the 
interest  of  $1  for  50  times  3  mo.,  or  150  mo. ;  and  the  interest  of  $1 
for  150  mo.  is  the  same  as  the  interest  of  $100  for  j^^  of  150  mo., 
or  1|  mo. 

6.  If  I  lend  $200  for  3  mo.,  for  how  long  a  time  should 
I  have  the  use  of  $50  to  balance  the  favor  ? 

7.  If  A  borrows  of  B  $1000  for  3  mo.,  what  sum 
should  A  lend  B  for  9  mo.  to  discharge  the  obligation  ? 

*733.  Peinciple. — The  interest  arid  rate  remaining  the 
samey  the  greater  the  principal  the  less  the  time,  and  the 
less  the  principal  the  greater  the  time. 

wit  ITT  EK     EXDItCISBS, 

733.  Find  the  average  term  of  credit 

1.  Of  $300  due  in  cash,  $500  due  in  3  mo.,  $750  due 
in  8  mo.,  and  $950  due  in  10  mo. 

OPERATION  Analysis— On  $300,  the  first 

3  0  0  X     0  =  0  payment,   there  is    no    interest, 

^^  ^  i^^nn  since  it  is  due  in  cash ;  the  int. 

b^^  X     d_15U0  ^^  ^5()Q  ^^^  3  ^^^  .g  ^j^^  ^^^  ^^ 

75  0  X      8  =  6000  the  int.  of  $1  for  1500  mo.;  the 

950  Xl0  =  9500  int.  of  $750  for  8  mo.  is  the  same 

^Kr\r\  vTtOOO  ^^  *^^^*  of  $1  for  6000  mo. ;  and 

^  the  int.  of  $950  for  10  mo.  is  the 

6 1-  mo.      same  as  the  int.  of  $1  for  9500  mo. 

Therefore,  the  whole  amt.  of  int. 

is  that  of  $1  for  1500  mo.  4-  6000  mo.  -f  9500  mo.,  or  17000  mo. ;  hut 

the  whole  debt  is  $2500  ;  and  the  int.  of  $1  for  17000  mo.  is  equal 

to  the  int.  of  $2500  for  ^^tttf  of  17000  mo.,  or  ^  mo. 

2.  Find  the  average  term  of  credit  of  $800  due  in  1  mo., 
$750  due  in  4  mo.^  and  $1000  due  in  6  mo. 


EQUATION     OF     PAYMENTS.  107 

Rule. — I.  Multiply  each  payment  ly  its  term  of  credit, 
and  divide  the  sum  of  the  products  iy  the  sura  of  the  pay- 
ments; the  quotient  is  the  average  term  of  credit. 

II.  (To  find  the  equated  time  of  payment,)  Add  the 
average  term  of  credit  to  the  date  at  which  the  several 
credits  begin. 

3.  On  the  first  day  of  December,  1876,  a  man  gave 
3  notes,  the  first  for  $500  payable  in  3  mo. ;  the  second 
for  $750  payable  in  6  mo.  ;  and  the  third  for  $1200  paya- 
ble in  9  mo.  What  was  the  average  term  of  credit,  and 
the  equated  time  of  payment  ? 

^  4.  Bought  merchandise  Jan.  1,  1875,  as  follows  :  $350 
on  2  mo.,  $500  on  3  mo.,  $700  on  6  mo.  What  is  the 
equated  time  of  payment  ? 

.  5.  A  person  owes  a  debt  of  $1680  due  in  8  months,  of 
which  he  pays  -J^  in  3  mo.,  J  in  5  mo.,  ^  in  6  mo.,  and 
■J^  in  7  mo.     When  is  the  remainder  due  ? 

6.  Bought  a  bill  of  goods,  amounting  to  $1500  on  6 
months'  credit.  At  the  end  of  2  mo.,  I  paid  $300  on 
account,  and  2  mo.  afterward,  paid  $400  on  account,  at 
the  same  time  giving  my  note  for  the  balance.  For  what 
time  was  the  note  drawn  ? 

OPEKATiON.  Analysis.  —  $300   paid 

300x4  =  1200  ^  ^^-  l>^fc)re  it  is  due,  and 

/lAAv.  o           Q(\f\  $400,   2  mo.  before  it  is 

, due,  are  equivalent  to  the 

800    '           )2000  use  of  $1  for  2000  months, 

21  or  the  use  of   $800  (the 

,^               ,         .      ^^               ,^  balance)  for  2^  mo.  bevond 

(6  mo.  -4  mo.)  +  2*  mo.=4i  mo.     ^he  original  time.     H;nce. 

the  note  was    drawn  for 
4i  mo.  after  the  second  payment. 


108  PERCENTAGE. 

7.  On  a  debt  of  $2500  due  in  8  mo.  from  Feb.  1,  the 
following  payments  were  made  :  May  1,  $250,  July  1, 
$300,  and  Sept.  1,  $500.     When  is  the  balance  due  ? 

8.  Find  the  average  term  of  credit,  and  the  equated 
time  of  payment  from  Dec.  15,  of  $225  due  in  35  da., 
$350  due  in  60  da.,  and  $750  due  in  90  da. 

9.  Dec.  1,  1874,  purchased  goods  to  the  amount  of 
$1200,  on  the  following  terms  :  25^  payable  in  cash, 
30^  in  3  mo.,  20^  in  4  mo.,  and  the  balance  in  6  mo. 
Find  the  equated  time  of  payment,  and  the  cash  value  of 
the  goods,  computing  discount  at  1%. 

734.  To  find  the  equated  time  when  the  terms  of 
credit  begin  at  different  dates. 

1.  J.  Prince  bought  goods  of  W.  Sloan  as  follows : 
June  1, 1874,  amounting  to  $350  on  2  mo.  credit ;  July  15, 
1874,  $400,  on  3  mo.  credit ;  Aug.  10,  $450,  on  4  mo. 
credit;  Sept.  12,  $600,  on  6  mo.  credit.  What  is  the 
equated  time  of  payment? 


OPERATION. 

$350 

due 

Aug.   1, 

350  X      0    =             0 

400 

«* 

Oct.  15, 

400  X    75    =      30000 

450 

<( 

Dec.  10, 

450  X  181     =      58950 

600 

(( 

Mar.  12, 

600  X  223     =    138800 
1800             1800)222750 
128f 

Hence  the  equated  time  is  124  da.  from  Aug  1,  or-  Dec.  3. 

Analysis. — Computing  the  terms  of  credit  from  Aug.  1,  the 
earliest  date  at  which  any  of  the  debts  become  due,  we  find  the 
terms  of  credit  to  be  from  Aug.  1  to  Oct.  15,  75  da. ;  to  Dec.  10, 
181  da.,  and  to  March  12,  228  da.  The  average  term  of  credit  is 
therefore  124  da.  from  Aug.  1,  and  the  equated  time  Dec.  3. 


EQUATION     OF     PAYMENTS.  109 

Proof. — Assume  as  the  standard  time  the  latest  date,  March  13, 
The  operation  will  then  be  as  follows  : 

350  X  223  =  78050 
400  X  148  =  69200 
450     X       92     =    41400 

600     X         0     = 0 

1800)178650 
99i 
Hence,  the  equated  time  is  99  da.  previous  to  March  12,  or  Dec.  S. 

2.  Peake  &  Co.  sell  to  Wm.  Jones  the  following  bills 
of  goods  :  March  1,  1875,  on  60  da.,  $800  ;  April  15,  on 
30  da.,  $350 ;  May  20,  on  4  mo.,  $3800. 

What  is  the  equated  time  for  settlement  ? 

Rule,— I.  Find  tJie  date  at  which  each  deit  becomes  due. 

II.  From  the  earliest  of  these  dates  as  a  standard  com^ 
pute  the  time  to  each  of  the  others, 

III.  Then  find  the  average  term  of  credit  and  equated 
time  as  in  (733). 

Proof. — Compute  the  terms  of  credit  backward  from  the 

latest  datCy  and  subtract  the  average  time  from  that  date 

for  the  equated  time. 

If  the  earliest  date  is  not  the  first  of  the  month,  it  is  more  con- 
venient to  assume  the  first  of  the  month  as  the  standard  date. 

3.  Bought  mdse.  as  follows  :  Jan.  15,  1876,  on  4  mo., 
$375  ;  Feb.  3,  on  60  da.,  $550 ;  March  25,  on  4  mo., 
$1100  ;  April  2,  on  30  da.,  $250.     Find  the  equated  time. 

4.  Ira  Blunt,  of  Gadsden,  Ala.,  bought  of  Opdyke  & 
Co.  the  following  bills  of  goods  on  4  months'  credit : 

Jan.  1,  1874,  $650  ;  Feb.  10,  $380 ;  March  12,  $900 ; 
March  18,  $350 ;  April  3,  $600. 

April  5,  he  discounted  his  bills  at  2^  per  month.  Find 
the  equated  time  of  payment,  and  the  discount. 


110 


0                                    PERi 

DENTAGE. 

5.  James  Smith 

to 

Thomas  Browk,  Dr. 

March  10,  1874. 

To  mdse. 

$835. 

"      18,     '' 

a 

(( 

330. 

"      26,     '' 

a 

a 

475. 

April      5,     '' 

(C 

66 

600. 

"       12,     '' 

cc 

66 

350. 

Allowing  30  days'  credit  on  each  of  the  bills,  what  i& 
the  equated  time  of  payment  ? 

6.  Purchased  goods  as  follows  : 

Sept.  15,  1875,  a  bill  of  $275,       on  3  mos. 
Oct.   10,     ''  "  351.50,  ''  60  da. 

"      28,     ''  ''  415.75,  "  30  da. 

NoY.    3,     "  "  500,        "    4  mos. 

Dec.  15,     "  "  710,        ''    3  mos. 

What  was  due  on  this  account  Aug.  10,  1876,  com- 
puting interest  at  7^  ? 

7.  I  have  four  notes,  as  follows  :  the  first  for  $425,  due 
April  1,  1875  ;  the  second  for  1615,  due  May  10,  1875  ; 
the  third  for  $1500,  due  May  28,  1875  ;  and  the  fourth 
for  $750,  due  June  10,  1875. 

At  what  date  should  a  single  note  be  made  payable,  to 
be  given  in  exchange  for  the  four  notes  ? 

ayeragi:ng  accounts. 

735.  An  Account  is  a  written  statement  of  debit 
and  credit  transactions,  with  their  respective  dates. 

B^it  means  wliat  is  owed  by  the  person  with  whom  the  account 
is  kept ;  credit,  what  is  due  to  him  from  the  person  keeping  the 
account. 

736.  To  Average  an  Account  is  to  lind,  either 


AVERAGIlJfG     ACCOUNTS. 


Ill 


the  equated  time  of  paying  the  balance,  or  the  cash  balance 
at  any  given  time. 

Each  item  of  a  book  account  should  draw  interest  from  the  time 
it  becomes  due. 


WRITTEN    EX  EMCISES  . 

737.  1.  Find  the  equated  time  of  paying  the  balance 
of  the  following  account. 
Dr.  William  Sampson.  Cr. 


1875. 

1875. 

Jan.  11 

To  mdse.   ,     .     . 

$750 

Feb.  10 

By  draft  at  60  da. 

$500 

Feb.    1 

**      "     at  3  mo. 

600 

Mar.  3 

*'  cash      .    .     . 

700 

Mar.  15 

**      **     at  6  mo. 

1500 

Apr.  15 

it        ii 

300 

May    3 

**      *'     at  4  mo. 

900 

Operation  I.    {Method  hy  Products) 


Due. 
Jan.   11. 
May     1. 
Sept.  15. 

**      3. 


Amt.     Days.       Product.        Paid. 


750  X  10  =       7500 

600  X  120  =    72000 

1500  X  257  =  3^5500 

900  X  245  =  220500 

3750  685500 

1500  125400 


Apr.  14. 
Mar.  3. 
Apr.  15. 


Amt. 
500  : 
700  : 
300  : 

1500 


103  =     51500 
61  =    42700 

104  =     31200 

125400 


2250 


) 560100 


248ff ,  or  249  da. 
Balance  due  249  da.  from  Jan.  1,  or  Sept,  7. 

Analysis.— Assuming  for  convenience  Jan.  1  as  the  standard 
date,  we  find  as  in  734  the  term  of  credit  of  each  debit  amount ; 
and,  reckoning  from  the  same  date,  the  time  to  each  credit  amount. 
Multiplying  each  amount  by  its  time  in  days,  and  adding  the  debit 
and  credit  products,  we  find  the  number  of  days'  interest  of  $1  due 
to  the  debtor,  and  the  number  of  days'  interest  of  $1  he  has  already 
received.  The  difference,  560100,  shows  the  number  of  days'  inter- 
est of  $1  still  due,  and  as  the  balance  is  $2250,  the  time  must  be 
^^  of  560100  da.,  or  249  da.  Hence,  the  equated  time  is  249  da. 
from  Jan.  1,  or. Sept,  7. 


il2 


PERCENTAGE. 


Operation  II.    {^Method  by  Interest.) 


Dt. 
$750  to  Jan,  11  (from  Jan.  1)= 

600  ''  Feb.   1  +  3  mo.  ,     =  4  mo. 
1500  ''  Mar.  15  +  6  mo.        =8  mo.  14  da., 

900  ''  May   3  +  4  mo.        =  8  mo.    2  da., 

$3750 

Gr, 
$500  to  Feb.  10  +  63  da.  =  3  mo.  13  da.,  int.  at  1%  per  mo.  $17.17 


10  da.,  int.  at  1%  per  mo.  $2.50 
24.00 
127.00 
"  72.60 

$226.10 


700  ''  Mar.    3 
300  *' Apr.15 


=  2  mo.    2  da., 
=  3  mo.  14  da.. 


14.47 
10.40 
$42.04 


$1500 

$226.10  -  $42.04  =  $184.06,  int.  at  1%  per  mo.  due. 
Int.  of  balance,  $2250,  for  1  mo.,  at  1^  =  $22.50. 
Hence,  $184.06  -i-  $22.50  =  8.18+  mo.,  or  8  mo.  6  da. 
8  mo.  6  da.  from  Jan.  1,  or  ISept.  7,  Equated  Time. 

In  this  operation,  12  %  per  annum  or  1  %  per  mo.  is  assumed  for 
convenience  ;  since  the  int.  at  1  ^/o  per  mo.  is  as  many  hundredths 
as  there  are  months,  and  one-third  as  many  thousandths  as  there 
are  days.  Thus,  the  int.  of  $249  for  2  mo.  9  da.  is  $498  +  $.747 
=  $5,727(571). 


2.  Find  the  equated  time  of  the  following  ; 
Dr,  William  Simpsok. 


(7n 


1874. 

1874. 

Aug.  5 

To  mdse.  at  3  mo. 

$720 

Oct.  10 

By  cash  .  .  . 

$500 

Sept.  10 

,.   -   -  2  ** 

850 

Dec.  15 

"  draft  at  60  da. 

450 

Nov.  3 

<(   f( 

1200 

''  25 

''  cash  .  .  . 

900 

1875. 

1875. 

Jan.  20 

''  sundr's  at  5  mo. 

620 

Jan.  3 

(t     <( 

250 

EuLE  1. — I.  Find  the  date  at  which  each  debit  item  is 
due,  and  each  credit  item  is  paid  or  due. 

II.  Tahe  the  first  day  of  the  month  in  the  earliest  date 
on  either  side  of  the  account  as  a  standard  date,  and  7nul' 


AVEBAGIKG     ACCOUKTS. 


113 


tiply  each  sum  due  or  paid  hy  the  number  of  days  between 
its  time  and  the  standard  date. 

III.  Add  the  products,  and  their  difference  divided  by 
the  balance  due  will  give  the  number  of  days  between  the 
standard  date  and  the  equated  time.     Or, 

Rule  2. — Find  the  time  of  each  item  from  the  standard 
date  as  before^  and  compute  the  interest  on  each  at  1%  a 
month.  The  differ e7ice  between  the  amount  of  interest  on 
each  side  divided  by  the  interest  of  the  balance  at  l%for 
one  month  will  be  the  equated  time. 

When  the  terms  of  credit  are  long,  Rule  2.  gives  the  shorter 
method. 

3.  Find  the  equated  time  of  the  following,  allowing 

^60  da.  credit  on  each  debit  item : 

Dr.  John  Deiscoll.  Cr. 


1877. 

1877. 

June  1 

To  mdse.  .  . 

$950 

Aug.  1 

By  cash   .  . 

$700 

July  6 

((      i( 

300 

Sept.20 

((                K 

1000 

Sept.  8 

it      t( 

1900 

Nov.  1 

it               it 

1200 

Oct.  20 

*'   '*   .  . 

2600 

H 


4.  What  is  the  equated  time  for  the  payment  of  the 
balance  of  the  following  account,  allowing  4  months' 
credit  on  all  the  debit  items  ? 

Dr.  DoDD,  Brown  &  Co.  Cr. 


1878. 

1878. 

Jan.  20 

To  mdse.  .  . 

$570 

Feb.  14 

By  mdse.  .  . 

$490 

'*  28 

300 

Mar.  1 

*'  cash   .  . 

1000 

Feb.  11 

720 

Apr.  2 

((  tt 

1800 

''    26 

835 

Mar.  10 

1150 

''    28 

930 

Apr.  15 

475 

114 


PERCEKTAGE. 


738.  1.  Find  the  cash  halance  of  the  following  account 
on  the  22d  of  August,  allowing  interest  at  6^  : 

Dr.  George  Hammond.  Cr, 


1875. 

1875. 

Mar.  15 

Tomdse.,at3mo. 

$600 

May  10 

By  cash      .    . 

$300 

Apr.    3 

<t      "      *'4ino. 

700 

July   1 

ti      ti 

400 

May  10 

^'      '*      '^6mo. 

1000 

Aug.l5 

{(      it 

500 

Operation. — By  averaging  the  account,  the  equated  time  for 
paying  the  balance,  $1100,  is  found  to  be  ^ov.  4.     (734.) 

True  present  worth  of  $1100  for  74  da.  (from  Aug.  22  to  Nov.  4) 
is  $1086.60,  or  cash  balance  Aug.  22. 
Or,  by  Interest  Method,  as  follows  : 

Dr. 
Int.   of    $600,  from  June  15  to  Aug.  22,    68  da.,      $6.71  (574.; 
**       **      700,     ''     Aug.    3  ''  "         19  da.,        2.19 

$8.90 
Cr. 
Int.  of  $1000,  from  Aug.  22  to  Nov.  10,    80  da.,    $13.15 


300,     *'     May  10  ''  Aug.  22, 104  da  , 
400,     ''     July    1'*  "         52  da., 

500,     "     Aug.  15  "  **  7  da.. 


513 
3.42 

58 

$22.28 
8.90 


Balance  of  interest  due  Hammond,        $13.38 
$1100  -  $13.38  =  $1086.62,  Cash  Balance,  Aug.  22. 

Analysis. — Charge  Hammond  with  interest  on  each  debit  item 
from  the  time  it  is  due  to  date  of  settlement,  and  credit  him  with 
interest  on  each  sum  paid  from  the  date  of  payment  to  date  of  set- 
tlement, also  on  each  debit  item  which  becomes  due  after  the  date 
of  settlement.  Hence,  he  is  entitled  to  interest  on  $1000  from 
Aug.  22  to  Nov.  10.  As  the  balance  of  interest  is  in  favor  of  Ham- 
mond, it  must  be  deducted  from  the  balance  of  the  account,  to  ob- 
tain the  cash  balance.  There  is  a  slight  difference  in  the  results, 
but  the  interest  method  is  the  more  accurate.  By  the  use  of  Inter- 
est Tables,  it  is  also  the  shorter  of  the  two  methods. 


AVERAGING     ACCOUNTS, 


115 


EuLE  1. — I.  Average  the  account^  and  find  the  equated 
time  of  payment  of  the  balance, 

11.  If  the  date  of  settlement  is  prior  to  the  equated  time, 
find  the  present  worth  of  the  balance  of  account  for  the 
cash  balance ;  if  later,  find  the  interest  of  the  balance  of 
account  for  the  intervening  time,  and  add  it  to  find  the 
cash  balance,    Ov, 

B>VL^  2.— Find  the  interest  on  each  debit  and  credit 
item,  from  the  time  it  is  due  or  paid  to  the  date  of  settle- 
me7it,  placing  on  the  same  side  of  the  account  the  interest 
on  each  item  due  prior  to  the  date  of  settlement,  and  on  the 
opposite  side  the  interest  on  each  item  due  after  the  date 
of  settlement.  If  the  balance  of  interest  is  on  the  same  side 
as  the  balance  of  the  account,  add  it,  if  on  the  other  side 
subtract  it ;  and  the  result  ivill  be  the  cash  balance  at  the 
date  of  settlement, 

2.  I  owe  $1500  duo  May  1,  and  $750  due  Aug.  15.  If 
I  give  my  note  at  30  da.  for  $450,  June  1,  and  pay  $370 
in  cash  July  15,  what  is  the  equated  time  for  paying  the 
balance  ;  and  what  would  be  due  in  cash  Dec.  10,  allow- 
ing interest  at  7^? 

3.  When  is  the  balance  of  the  following  account  due 
per  average  ? 

Dr.  0.  B.  TiMPSOiT.  Cr. 


1875. 

1875. 

Aug.  10 

To  mdse.  @  60  da.  . 

$751.35 

Oct.    3 

By  cash 

$300.00 

Sept.  5 

"      "      @30da.  . 

425.00 

Nov.  15 

"   note  @  90  da.    . 

450.00 

Nov.    1 

"      ''      @90da.  . 

927.83 

Dec.  20 

'*   cash     .... 

500.00 

Dec.    5 

"      "      @30da.  . 

1200.00 

116 


PERCENTAGE. 


4.  What  IS  the  cash  balance  of  the  aboye  account  Jan.  1, 
1876,  allowing  interest  at  10^? 

5.  Find  the  equated  time,  and  cash  balance  July  1,  of 
the  following,  allowing  7^  interest : 


Dr. 


Thomas  Smith. 


Cr. 


Jan.    4 

To  mdse.  @  4  mo. 

$1600 

Feb.    1 

By  mdse.  @  4  mo. 

$500 

"      6 

"       "      @3mo. 

1500 

Mar.   2 

"  cash    .     .     . 

2000 

Apr.  10 

"      "      @60da. 

3000 

"     25 

((     (I 

3150 

"    28 

"      "      @30da. 

2500 

Apr.  16 

ti.    (( 

800 

6.  Average  the  following  account,  and  find  for  what 
amount  a  note  at  60  days  should  be  given  Aug.  1,  to  pay 
the  balance,  interest  at  6^. 

Dr.  Orson  Hinman.  Or. 


1875. 

1875. 

«♦ 

Apr.   2 

To  charges 

$87.25 

Feb.  25 

By  mdse.  @  8  mo. 

$600 

May  15 

((        (i 

35.75 

Mar.    3 

*'      «      @6  '' 

300 

Apr.    1 

''      "      @6  '' 

500 

739.  1.  Average  the  following  Account  Sales,  and  find 
when  the  net  proceeds  are  due.     (543.) 

Account  Sales  of  1200  iils.  of  flour  received  from 
SmitJi,  Tyler  &  Co.,  Cincinnati. 


Date. 

Buyer. 

Quantity. 

Price. 

Amount. 

1876. 
May  1 
June  5 

'*  15 
July   1 

J.  Brooke 
W.  Long 
A,  Bruce 
W.  Case 

300  bbl. 
450   '' 
250   '' 
200  " 

%  $5.50,  3  mo. 
%    6.20,  4  mo. 
%    6.50,  6  mo. 
@    5.75,  2  mo. 

$1650.00 
2790.00 
1625.00 
1150.00 

$721500 


AVERAGIKG     ACCOUKTS.  117 

Charges. 

Apr.  28.    Freight     . $674.50 

*'      "      Cartage 37.50 

May    1.     Storage 191.00 

Commission  on  $7215,  @  2}  fc     .    .    .    162.34 

Total  charges $1065.34 

Net  proceeds  due  per  average $6149.66 

OPERATION. 

'  -Average  of  sales,  found  by  the  method  of  Equation  of  Payments, 
Oct.  ly  which  is  the  date  at  which  the  commission  is  due. 

Average  of  charges,  including  commission  (Oct.  1),  May  22. 
Equated  time  of  $7215  due  Oct.  1,  and  $1065.34 due  May  22,  0cL2Ji,, 
date  when  the  net  proceeds  are  due. 

EuLE. — I.  Average  the  sales  alone^  and  the  result  will 
ie  the  date  to  be  given  to  the  commission  and  guaranty. 

II.  Make  the  sales  the  credits  and  the  charges  the  debits, 
and  find  the  equated  time  for  paying  the  balance. 

2.  Make  an  account  sales,  and  find  the  net  proceeds 
and  the  time  the  balance  is  due  : 

Wm.  Brown,  of  N.  York,  sold  on  acct.  of  J.  Berry,  of  Chi- 
cago, June  1,  350  bu.  Winter  Wheat,  @  $1.35,  at  60  da. ; 
June  15,  275  bu.  Spring  Wheat,  @  $1. 75,  at  90  da. ;  July  3, 
1260  bu.  Indian  Corn,  @  $.79,  at  6  mo.;  and  July  10, 
375  bu.  Eye,  @  $1.02,  at  3  mo.  Paid  freight,  May  28, 
$567.50;  cartage.  May  30,  $22.50;  insurance,  June  5, 
$56.25  ;  and  charged  com.  at  3^%,  and  1^%  for  guaranty. 

3.  Sold  on  account  of  Brown,  Sampson  &  Co.,  at  6 
mo. :  Oct.  1,  1874,  13  hhd.  sugar,  averaging  1520  lb.,  (^ 
$.12^  ;  Oct.  5,  15  chests  Hyson  Tea,  each  95  lb.,  @  $1j05. 
Paid  charges :  Oct.  3,  Insurance,  $85  ;  Oct.  10,  Cooper- 
age, etc.,  $24.50  ;  Oct.  20,  Cartage,  $125.  Charged  com- 
mission and  guaranty,  4:^%.  Make  an  account  sales,  and 
find  the  equated  time  for  paying  the  net  proceeds. 


118 


PERCENTAGE. 


740. 


SYNOPSIS    FOR    REVIEW. 

Exchange,  3.  Domestic  Exchange,  3.  i^(?r- 
ej^^  Exchange.  4.  ^2^^  <?/  Exchange.  5. 
/S'^^  <?/  Exchange.  6.  /Sji^A^  l>ra/if  <?r  -Si7^. 
7.  2Vm^  Draft  or  Bill.  8.  Buyer  or  Be- 
mitter.  ^.Acceptance.  10.  Par  of  Exchange, 
11.  Course  or  Bate  of  Exchange. 

A  Sight  Draft.    2.  Set  of  Exchange. 

TOO.  i  rp    fi  ^  t  Cost  of  Draft.  Formvla. 

701.  i      ^  \  Face  of  Draft. 


1. 

Defs.  . 

2. 

Forms. 

3. 

Inland 

Exch. 

j  1.  TOO.  ) 
( 2.  701.  ) 


4  Foreign 
Exch'ge. 


5.  Arbitra- 
tion of 
Exch'ge. 


22.  Custom- 
house 
Business. 


23.    Equation 
OF  Paym'ts. 


24    AVEKAGINa 

Accounts. 


j  1.  Money  of  Account, 


1.  Defs.   . 
Sterling  Bills,  or  Exchange. 

2.  Exchange  with  Europe — how  effected. 


f:l 


To  find 


j  CostofBiU. 
\  Face  of  Bill. 


1.  Defs. 


706. 
707. 

1.  Arbitration  of  ExcJiange. 

2.  Simple  Arbitration. 
I  3.  Compound  Arbitration. 

Rule,  I,  II,  m,  IV. 

1.  Custom  Rouse.  2.  Port  of 
Entry.  3.  Collector.  4. 
Clearance.  5.  Manifest.  6. 
Duties  or  Customs.  7.  Tariff. 
S.  Specific  Duty.  9.  Ad  Val- 
orem Duly.  10.  Gross  Wght. 
11.  Net  Weight- 
To  find  the  Duty. 
1.  Equation  of  Payments.  2. 
Equated  Time.  3.  Term  of 
Credit.  4  Average  Term  of 
Credit. 

2.  Principle. 

3.  733.    Rule,  I,  IL 

4  734.     Rule,  I,  II,  III.     Proof. 

1.  Defs.    1.  Account.  2.  To  Average  an  AceL 

2.  737.    Rule  1,  I,  II,  III.    Rule  2. 

3.  738.    Rule  1, 1,  II.    Rule  2. 
4  739.    Rule,  I.  II. 


1.  Defa.  . 


2.  726. 


1.  Defs.  < 


ORJLL      EXERCISES. 

741.  1.  A  father  is  30  years  old,  and  his  son  6  ;  ho\v 
many  times  as  old  as  the  son  is  the  father  ? 

2.  30  are  how  many  times  6  ?    30  ~-  6  ==  ? 

3.  What  part  of  $30  are  $6  ?    Of  20  cents  are  5  cents  ? 

4.  What  is  the  relation  of  8  to  2  ?     Of  40  rd.  to  4  rd.  ? 

5.  What  relation  has  12  to  3  ?     60  lb.  to  20  lb.  ? 
Compai-e  the  following,  and  give  their  relative  values. 


6.  75  with  5. 

7.  25  with  Gf 

8.  1  with  7. 


9.     \  with  7. 

10.  ^  with  3f 

11.  .9  with  .3. 


12.  $.G  with  $.2. 

13.  .42  with  .3. 

14.  f  with  f . 


DEFINITIONS. 

743.  Ratio  is  the  relation  between  two  numbers  of 
the  same  unit  value,  expressed  by  the  quotient  of  the  first 
divided  by  the  second.  Thus  the  ratio  of  12  to  4  is 
12  ^  4  =  3. 

743.  The  Sign  of  ratio  is  the  colon  ( : ),  or  the  sign 

of  division  with  the  line  omitted. 

Thus,  the  ratio  of  9  to  3  is  expressed  9:3,  or  9^3,  or  in  the  form 
of  a  fraction  |,  and  is  read,  the  ratio  of  9  to  3,  or  9  divided  by  3. 

744.  The  Terms  of  a  ratio  are  the  two  numbers 
compared. 

745.  The  Antecedent  is  the  first  term,  or  dividend. 

746.  The  Consequent  is  the  second  term,  or  divisor. 


120  RATIO. 

747.  The  Value  of  a  ratio  is  the  quotient  of  the  antece- 
dent divided  by  the  consequent,  and  is  an  abstract  number. 

Thus,  in  the  ratio  $18  :  $6,  $18  and  $6  are  the  terms  of  the  ratio  ; 
$18  is  the  antecedent ;  $6  is  the  consequent ;  and  3,  the  quotient  of 
$18  H-  $6,  is  the  value  of  the  ratio. 

748.  A  Simple  Ratio  is  the  ratio  of  two  numbers ; 
as  10 :  5. 

749.  A  Compound  Ratio  is  the  ratio  of  tho 
products  of  the  corresponding  terms  of  two  or  more  sim^ 
pie  ratios. 

Thus  the  ratio  compounded  of  the  simple  ratios, 
I  ;  ^^  f  may  be  expressed  {  ^^^  ^^  \^^ ":  J'j  /|^  }  =72  :  48  ; 
Or,  f  X  V^  =  I  =  3  :  2. 

When  the  multiplication  is  performed  the  result  is  a  simple  ratio. 

750.  The  Reciprocal  of  a  ratio  is  1  divided  by  the 
ratio  (196),  or  it  is  the  consequent  divided  by  the  ante- 
cedent. Thus  the  ratio  of  8  to  9  is  8  :  9,  or  f ,  and  its 
reciprocal  is  f. 

The  ratio  of  two  fractions  is  obtained  by  reducing  them  to  a 
common  denominator,  when  they  are  to  each  other  as  their  nume- 
rators (241). 

If  the  terms  of  a  ratio  are  denominate  numbers,  they  must  b« 
reduced  to  the  same  unit  value. 

751.  From  the  preceding  definitions  and  illustrations 
are  deduced  the  following 

Formulas. — 1.   The  Ratio  =  Antecedent -r-  Consequent. 

2.  The  Consequent  ==  Antecedent-^ Ratio. 

3.  The  Afitecedent  =  Consequent  x  Ratio. 


RATIO.  121 

753.  Since  the  antecedent  is  a  dividend,  and  the  con- 
sequent a  divisor,  any  change  in  either  or  both  of  the 
terms  of  a  ratio  will  aflfect  its  value  according  to  the  laws 
of  division  or  of  fractions  (200),  which  laws  become  the 

Gei^eral  Principles  of  Eatio. 

1.  Multiply inq  the  antecedent y  or  ),..,,.  ,.     ,,       ,. 

^.  .;.      ,^  ^  }  Multiplies  the  ratio. 

Dividing  the  consequent,  ) 

2.  Dividinq  the  antecedent,  or        )  r^  •  •  7     ,i        ,- 
,^  ^,.  /.      ,^  ",        }  Divides  the  ratio. 
Multiplying  the  consequent,       ) 

3.  Multiplyinq  or  dividing   loth  ]  ^  ,   ,  ,, 

^  ^ -,  ^ ,      -,  _Li    [  Does  not  chanqe  the 

antecedent  and  consequent  by  Y  , . 

the  same  number,  ) 

753.  These  principles  may  be  embraced  in  one 

GENERAL  LAW. 

A  change  in  the  antecedent  produces  a  like  change  in 
the  ratio ;  hut  a  change  in  the  consequent  produces  an 
OPPOSITE  change  iii  the  ratio. 

JSXERCISES. 

754.  1.  Express  the  ratio  of  11  to  4  ;  of  16  to  2  ;  of  20 

lo  6| ;  of  $36  to  112  ;  of  9  lb.  to  27  lb.  ;  of  ^  bu.  to  9  bu. 

2.  Can  you  express  the  ratio  between  $15  and  5  lb.  ? 
Why  not  ? 

3.  Indicate  the  ratio  of  18  to  20  in  two  forms.  What 
are  the  terms  of  the  ratio  ?  The  antecedent  ?  The  co^^- 
sequent  ?    The  hind  of  ratio  ?    The  value  of  the  ratio. 

In  like  manner  express,  analyze,  and  give  the  value, 

4.  Of  80  to  120  ;  of  12^  to  37i  ;  of  l^  to  |. 

2x27x42 


5.  Of  5.2  to  1.3;  of  f  to^;  of 


12x4x126' 


122  RATIO. 

6.  The  antecedents  of  a  ratio  are  7  and  10,  and  the 
consequents,  5  and  4.    What  is  the  value  of  the  ratio  ? 

7.  The  first  terms  of  a  ratio  are  18,  12,  and  30,  the 
second,  54,  6,  and  15.  What  is  the  kind  of  ratio  ?  Ex- 
press in  three  forms.     Find  its  value  in  the  lowest  terms. 

Solve,  and  state  the  formula  applied  to  the  following  : 

8.  The  consequent  is  3;^,  the  antecedent  -^f ;  what  is 
the  ratio  ? 

9.  The  antecedent  is  60,  the  ratio  7  ;  what  is  the  con- 
sequent ? 

10.  The  consequent  is  $6.12^,  the  ratio  ^ ;  what  is 
the  antecedent  ? 

11.  The  ratio  is  2f,  the  antecedent  i^  of  | ;  what  is  the 
consequent  ? 

12.  The  ratio  is  6,  the  consequent  1  wk.  3  da.  12  hr. ; 
what  is  the  antecedent? 

13.  Express  the  ratio  of  120  to  80,  and  give  its  value 
in  the  lowest  terms. 

14.  Make  such  changes  in  the  last  example  as  will 
illustrate  Prik.  1. 

15.  With  the  same  example,  illustrate  Prin.  2. 

16.  Illustrate  by  the  same  example  Prin.  3. 

17.  Find  the  reciprocal  of  the  ratio  of  75  to  15. 

18.  Find  the  reciprocal  of  the  ratio  of  2  qt.  1  pt.  to 
4  gal.  1  qt.  1  pt. 

What  is  the  ratio 

19.  Of  40  bu.  4.5  pk.  to  25  bu.  2  pk.  1  qt. 

20.  Of  6  A.  110  P.  to  10  A.  60  P. 

21.  Of  25  lb.  11  oz.  4  pwt.  to  19  lb.  5  oz.  8  pwt. 

2..  Ofl?itoi^a 


ORAL      EXBMC  IS  E8, 

755.  1.  What  is  the  ratio  of  4  to  2  ?     Of  6  to  1  ?     Of 

14  to  7  ?    Of  21  to  3  ? 

2.  Find  two  numbers  that  have  the  same  quotient  as 
8^2.     As  27  -r-  3.    As  16  -r-  4.     As  30  -^  6.     As  4-^|. 

3.  Express  in  the  form  of  a  fraction  the  ratio  of  26  to 
13.     Of  32  to  8. 

4.  Express  in  both  forms  the  ratio  of  two  other  num- 
bers equal  to  the  ratio  of  10  to  2.    Of  15  to  5.    Of  12  to  3. 

5.  If  4  stamps  cost  12  cents,  what  will  20  stamps  cost 
at  the  same  rate  ? 

6.  What  number  divided  by  12,  gives  the  same  quo- 
tient as  20  -^  4  ? 

7.  What  number  has  the  same  ratio  to  12,  that  20  has 
to  4? 

8.  To  what  number  has  48  the  same  ratio  that  80  has 
to  5  ?    That  24  has  to  3  ? 

9.  The  ratio  of  20  to  5  is  the  same  as  the  ratio  of  what 
number  to  4  ?     To  6  ?     To  5^  ?     To  6^  ? 

10.  The  ratio  of  45  to  9  is  the  same  as  the  ratio  of  15 
to  what  number  ?     Of  30  to  what  number  ? 

11.  28  is  to  7  as  8  is  to  what  number  ? 

12.  56  is  to  8  as  what  number  is  to  5  ? 

13.  63  -T-  what  number  equals  the  ratio  of  36  to  4? 


124  PKOPORTIOK. 

DEFINITIONS. 

756.  A  Proportion  is  an  equation  in  which  each 
member  is  a  ratio  ;  or  it  is  an  equality  of  ratios. 

757.  The  equality  of  the  two  ratios  may  be  indicated 
by  the  sign  =  or  by  the  double  colon  : : 

Thus,  we  may  indicate  that  the  ratio  of  8  to  4  is  equal  to  that  of 
6  to  3,  in  any  of  the  following  ways  : 

8:4rr6:3,  8:4::6:3,         |  =  |         8^-4  =  6^3. 

This  proportion,  in  any  of  its  forms,  is  read.  The  ratio  of  8  to  4  is 
equal  to  the  ratio  of  6  to  3,  or,  8  is  to  4  as  6  is  to  3, 

758.  Since  each  ratio  consists  of  two  terms,  every  pro- 
portion must  consist  of  at  least /o^^r  terms.  Each  ratio  is 
called  a  Couplet,  and  each  term  is  called  a  Proportional. 

759.  The  Antecedents  of  a  proportion  are  the  first 
and  third  terms,  that  is,  the  antecedents  of  its  ratios. 

760.  The  Consequents  are  the  second  and  fourth 
terms,  or  the  consequents  of  its  ratios. 

761.  The  JExtremes  are  the  first  and  fourth  terms. 
763.  The  Means  are  the  second  and  third  terms. 

In  the  proportion  8  :  4  : :  10  :  5,  8,  4,  10,  and  5  are  the  propor- 
tionals; 8  :  4  is  the  first  couplet,  10  :  5  the  second  couplet ;  8  and  10 
are  the  antecedents,  4  and  5  are  the  consequents;  8  and  5  are  the  ex- 
tremes, 4  and  10  are  the  means. 

Three  numbers  are  proportional,  when  the  ratio  of  the  first  to  the 
second  is  equal  to  the  ratio  of  the  second  to  the  third.  Thus  the 
numbers  4,  10,  and  25  are  proportional,  since  4 :  10  =  10  :  25,  the 
ratio  of  each  couplet  being  f , 

When  three  numbers  are  proportional,  the  second  tenn  is  called 
a  Mean  Proportional  between  the  other  two. 


PROPORTION.  125 

The  proportion    8  :  4  : :  10  :  5    may  be  expressed  thus,  i=^ 

(757).     Reducing  these  fractions  to  equivalent  ones  having  a  com- 

.     ^       8x5      10x4 
mon  denominator,  -— —  =  . 

Since  these  fractions  are  equal,  and  have  a  common  denominator, 
their  numerators  are  equal,  or  8  x  5  =  10  x  4. 

763.  Principles. — 1.   The  product  of  the  extremes  of 
'  a  proportion  is  equal  to  the  product  of  the  means, 

2.  The  product  of  the  extremes  divided  iy  either  mean 
will  give  the  other  mean, 

3.  Tlie  product  of  the  means  divided  iy  either  extreme 
will  give  the  other  extreme. 

EXJERCISJES. 

764.  1.  The  ratio  of  4  to  10  is  equal  to  the  ratio  of  6 
to  15.     Express  the  proportion  in  all  its  forms  (757). 

Brill  Exercise. — How  many  terms  has  a  proportion  ?  What  are 
they  called  ?    How  many  ratios  ?     What  are  they  called  ? 

Name  the  proportionals  in  example  1 ;  the  couplets ;  the  ante- 
cedents ;  the  consequents  ;  the  extremes  ;  the  means.  What  is  the 
product  of  the  extremes  ?  Of  the  means  ?  What  is  the  dividend 
of  the  first  ratio  ?  The  divisor  of  the  second  ratio  ?  The  divisor 
of  the  first  ratio  ?  The  dividend  of  the  second  ratio  ?  In  the  frac- 
tional form  what  are  the  numerators  ?    The  denominators  ? 

2.  The  ratio  of  6  to  15  equals  the  ratio  of  8  to  20. 

3.  The  ratio  of  4|^  to  18  equals  the  ratio  of  6  to  24. 
Change  to  the  form  of  equations  by  Prin.  1  : 


4.  12  :  1728  : :  1  :  144. 

5.  2|  :  17  : :  20  :  143^. 


6.  27.03  :  9.01  : :  16.05  :  5.35. 

7.  f  :f  ::|:^. 

8.  The  extremes  are  15  and  48,  and  one  of  the  means 
is  10.     Find  the  other  mean. 

9.  The  means  are  25  and  75,  and  one  of  the  extremes 
is  12^.     Find  the  other  extreme. 


126 


p  R  o  p  o  R  T I  o  :n^  . 


The  required  or  omitted  term  in  a  proportion,  or  in  an  operation, 
will  hereafter  be  represented  by  x. 

Find  the  term  omitted  in  each  of  the  following  pro- 
portions : 

IT.  4|yd.:cryd.::$9|:  $27.25. 

18.  x:  0.01  ::  16.05:  5.35. 

19.  |yd.::?ryd.::$|:  $59.0625. 

20.  -^:|::^:|. 

21.  .r:38i::8|:76f 

22.  7.5:18::a:oz.  :  7^V  ^z- 


11.  8:52::20:ir. 

12.  12  :ic::  1:144 

13.  a;:  20::  120:  50. 

14.  $80:  $4::^:  4. 

15.  2.5:62.5::5:ir. 

16.  $175.35:  $2:  ::i:f. 


SIMPLE    PROPOETION. 

765.  A  Simple  JProjfortion  is  an  expression  of 
equality  between  two  simple  ratios.  It  is  used  to  solye 
problems  of  which  three  terms  are  given,  and  the  fourth 
is  required. 

Of  the  three  given  numbers,  two  mnst  always  be  of  the  same 
kind ;  and  the  third y  of  the  same  kind  as  the  required  term. 

766.  A  Statement  is  the  arrangement  of  these 
terms  in  the  form  of  a  proportion. 

WRITTEN    EXEItCISTSS. 

767.  1.  If  4  tons  of  coal  cost  $24,  what  will  be  the 
cost  of  12  tons  at  the  same  rate  ? 


STATEMENT. 

4T.:  12  T.  ::  $24:$a; 

OPERATION. 


12  X  24-^4=r$72 
Or  By  Cancellation. 

12  X  t^' 


>x^- 


$72 


Analysis. — Since  4  tons  and  12 
tons  have  the  same  unit  value,  they 
can  be  compared,  and  will  form  one 
couplet  of  the  proportion. 

For  the  same  reason  $24  the  cost 
of  4  tons,  and  %x  the  cost  of  12  tons, 
will  form  the  other  couplet. 

Then  by  Prin.  3,  $a;=  24  x  12  -5-4 
z=$72. 


PROPORTION.  127 

Proof. — 4  x  72=12  x  24.  (763,  Prin.  1.)  In  practice,  that  number 
which  is  of  the  same  unit  value  as  the  required  term,  is  generally 
made  the  antecedent  of  the  second  couplet  or  third  term  of  the  pro- 
portion, and  the  required  term,  ic,the  fourth  term.  The  terms  of  the 
first  couplet  are  so  arranged  as  to  have  the  same  ratio  to  each  other, 
as  the  terms  of  the  second  couplet,  have  to  each  other,  which  is 
easily  determined  by  inspection.  The  product  of  the  means  12  and 
24,  divided  by  the  given  extreme  4,  gives  the  other  extreme,  or 
required  term,  $72.    (763,  Prin.  3.) 

Drill  exercises  like  the  following,  will  soon  make  the  pupil 
familiar  with  the  principles  and  operations  of  proportion. 

2.  If  4  horses  eat  12  bushels  of  oats  in  a  given  time, 
how  many  bushels  will  20  horses  eat  in  the  same  time  ? 

In  this  example,  what  two  numbers  have  the  same  unit  value  ? 
What  do  they  form  ?  What  is  the  denomination  of  the  third  term  ? 
Of  the  required  term  ?  What  is  the  antecedent  of  the  second  « 
couplet  ?  From  the  conditions  of  the  question,  is  the  consequent 
of  the  second  couplet  or  required  term,  greater  or  less  than  the 
antecedent  ?  If  greater,  how  must  the  antecedent  and  consequent 
of  the  first  couplet  compare  with  each  other  ?  If  less,  how  com- 
pare ?  What  is  the  ratio  of  the  first  couplet  ?  Why  not  20  to  4  ? 
Make  the  statement.    How  is  the  required  term  found  ? 

3.  If  96  cords  of  wood  cost  1240,  what  will  40  cords  cost  ? 

4.  If  20  lb.  of  sugar  cost  $1.80,  find  the  cost  of  45  lb. 

5.  If  18  bu.  of  wheat  make  4  barrels  of  flour,  how  many 
barrels  will  200  bu.  make  ? 

EuLE. — I.  Mahe  the  statement  so  that  two  of  the  given 
numbers  which  are  of  the  same  unit  value,  shall  form  the 
first  couplet  of  the  proportion,  and  have  a  ratio  equal  to 
the  ratio  of  the  third  given  term  to  the  required  term. 

II.  Divide  the  product  of  the  means  iy  the  given  extreme^ 
and  the  quotient  will  he  the  number  required. 


128  PROPORTION. 

CAUSE    AND    EFFECT. 

768.  The  terms  of  a  proportion  have  not  only  the 
relations  of  magnitude^  but  also  the  relations  of  cause 
and  effect 

Every  problem  in  proportion  may  be  considered  as  a 
comparison  of  two  causes  and  two  effects. 

Thus,  if  4  tons  as  a  cause  will  bring  when  sold,  $24  as  an  effect, 
12  tons  as  a  cause  will  bring  $72  as  an  effe^^t.  Or,  if  6  horses  as  a 
cause  draw  10  tons  as  an  effect,  9  horses  as  a  cause  will  draw  15 
tons  as  an  effect, 

769.  Since  like  causes  produce  like  effects,  the  ratio 
of  two  like  causes  equals  the  ratio  of  two  like  effects  pro- 
duced by  these  causes.     Hence, 

1st  cause  :  2d  cause  : :  1st  effect :  2d  effect. 

WRITTEN    BXEUCISES. 

770.  1.  If  8  men  earn  $32  in  one  week,  how  much  will 
15  men  earn  at  the  same  rate,  in  the  same  time  ? 

STATEMENT.  ANALYSIS. — In  this  ex- 

ist cause.         2d  cause.       1st  effect  2d  effect     ample  an  eJf(3C^  is  required. 
8  men   :    15  men    ::    $32    :    ^x        The  first  cause  is  8  men, 

the  second  cause  15  men, 

OPERATION.  ,      .  ^,  T., 

and  since  they  are  like 


%X  z=:15  Xt3/^-r-o^^=^bO  causes  they  can  be  com- 

pared. 
The  effect  of  the  first  cause  is  $32  earned,  the  effect  of  the  second 
cause  is  %x  earned,  or  the  required  term.  Since  like  effects  have 
the  same  ratio  as  their  causes  (769),  the  causes  may  form  the 
first  couplet,  and  the  effects  the  second  couplet  of  the  proportion. 
The  required  term  is  readily  obtained  by  (7C>3,  3). 

2.  If  20  bushels  of  wheat  produce  6  barrels  of  flour, 
how  many  bushels  will  be  required  to  produce  24  barrels  ? 


PROPORTION.  129 

STATEMENT.  ANALY8IS.~In  this  ex- 

ist cause.     2d  cause.     1st  effect.     2d  effect.      ample  a  cause  is  required. 
20  bu.  :  iiJ  bu  : :  6  bbl.  :  24  bbl.        The  first  cause  is  20  bu., 

the  second  cause  is  x  bu. 

OPERATION.  ,,  .      , 

or  the  required  term. 

a;  bu.  =  20  X  24  -r-  6  =  80  bu.  The  effect  of  the  first 

cause  is  0  bbl.  of  flour, 
the  effect  of  the  second  cause  is  24  bbl.  of  flour.  Since  like  causes 
have  the  same  ratio  as  their  effects  (709),  the  statement  is  made 
as  in  Ex.  1,  and  the  required  term  found  by  (703,  2). 

3.  If  5  horses  consume  10  tons  of  hay  in  8  mo.,  how 

many  horses  will  consume  18  tons  in  the  same  time  ? 

Drill  Exercise. — In  this  example,  what  is  the  first  cause  ?  The 
second  cause  ?  The  first  effect  ?  The  second  effect  ?  Is  the  re- 
quired term  a  cause  or  an  effect  ?  A  mean  or  an  extreme  ?  What 
is  the  first  couplet  ?  What,  the  second  ?  Make  the  statement. 
How  is  the  required  term  found  ? 

4.  If  8  yards  of  cloth  cost  $6,  how  many  yards  can  be 
bought  for  $75  ? 

5.  How  many  men  will  be  required  to  build  32  rods  of 
wall  in  the  same  time  that  5  men  can  build  10  rods  ? 

Rule. — I.  Arrange  the  terms  in  the  statement  so  that 
the  ratio  of  the  causes  which  form  the  first  couplet,  shall 
equal  the  ratio  of  the  effects  which  form  the  second  couplet y 
putting  X  in  the  place  of  the  required  term. 

II.  If  the  required  term^  te  an  extreme,  divide  the  pro- 
duct of  the  means  by  the  given  extreme ;  if  the  required 
term  he  a  mean,  divide  the  product  of  the  extremes  hy  the 
given  mean. 

To  shorten  the  operation,  equal  factors  in  the  first  and  second,  or 
in  the  first  and  third  terms  may  be  canceled. 

Solve  the  following  by  either  of  the  foregoing  methods  : 

6.  If  5  sheep  can  be  bought  for  $20.75,  how  manj 
sheep  can  be  bought  for  $398.40  ? 


130  PEOPORTION. 

7.  When  10  barrels  of  flour  cost  $112.50,  what  will  be 
the  cost  of  476  barrels  of  flour  Z 

8.  If  a  railroad  train  run  30  miles  in  50  min.,  in  what 
time  will  it  run  260  miles  ? 

9.  How  many  bushels  of  peaches  can  be  purchased  for 
$454.40,  if  8  bushels  cost  $10.24  ? 

10.  If  a  horse  travel  12  miles  in  1  hr.  36  min.,  how  far, 
at  the  same  rate,  will  he  travel  in  15  hours  ? 

11.  How  many  days  will  12  men  require  to  do  a  piece 
of  work,  that  95  men  can  do  in  7^  days  ? 

12.  If  f  of  an  acre  of  land  cost  $60,  what  will  45|- acres 
cost? 

13.  At  the  rate  of  72  yards  for  £44  16s.,  how  many 
yards  of  cloth  can  be  bought  for  £5  12s.  ? 

14.  If  ^  of  a^barrel  of  cider  cost  $1^,  what  is  the  cost 
of  I  of  a  barrel? 

15.  If  the  annual  rent  of  35  A.  90  P.  is  $284.50,  how 
much  land  can  be  rented  for  $374.70  ? 

16.  What  will  87.5  yd.  of  cloth  cost,  if  If  yd.  cost  $1.26  ? 

17.  If  by  selling  $5000  worth  of  dry  goods,  a  merchant 
gains  $456.25,  what  amount  must  he  sell  to  gain  $1000  ? 

18.  Bought  coal  at  $4.48  per  long  ton,  and  sold  it  at 
$7.25  per  short  ton.     What  was  the  gain  per.  ton  ? 

19.  What  will  be  the  cost  of  a  pile  of  wood  80  ft.  long, 
4  ft.  wide,  4  ft.  high,  if  a  pile  18  ft.  long,  4  ft.  wide,  6  ft. 
high  cost  $30.24? 

20.  If  36  bu.  of  wheat  are  bought  for  $44.50,  and  sold 
for  $53.50,  what  is  gained  on  480  bu.  at  the  same  rate  ? 

21.  If  a  business  yield  $700  net  profits  in  1  yr.  8  mo.,  in 
what  time  will  the  same  business  yield  $1050  profits  ? 


PROPORTION. 


131 


COMPOUND    PROPOETION. 

771.  A  Compound  Proportion  is  an  expression 
of  equality  between  two  ratios,  one  or  both  of  which  are 
compound. 

AH  the  terms  of  every  problem  in  compound  proportion  appear 
in  couplets,  except  one,  and  this  is  always  of  the  same  unit  value  as 
the  required  term. 

The  order  of  the  ratios,  and  of  the  terms  composing  the  ratios,  is 
the  same  as  in  simple  proportion. 


WRITTEN     EXERCISES, 

773.  1.  If  18  men  build  126  rd.  of  wall  in  60  da., 
working  10  hr.  a  day,  how  many  rods  will  6  men  build  in 
110  da.,  working  12  hr.  a  day? 


STATEMENT. 

0 

r. 

18  men     :       Omen    ^       ^o'ds.  rods. 

X 

0 

60  days    :  110  days    V  : :  126  :  x 

^U 

11011 

10  hours  :     12  hours  ) 

5  00 

U 

nPT?,!?  ATTON" 

i0 

m^^ 

11               42 

5 

463 

r<^B:_0xM0Xl^xW_,«,_9,„     , 

93|=a;rd. 

^          rA^tiiAs^xA     -     i      y-iiva. 

$      5 

Analysis. — All  the  terms  in  this  example  appear  in  couplets,  ex- 
cept 126  rods,  which  is  of  the  same  unit  value  as  the  required  term, 
and  is  made  the  third  term  of  the  proportion,  and  x  rods,  the  fourth. 

The  required  number  of  rods  depends  upon  ^^7*66  conditions  :  1st, 
the  number  of  men  employed  ;  2d,  the  number  of  days  they  work  ; 
and  3d,  the  number  of  hours  they  work  each  day. 

Consider  each  condition  separately,  and  arrange  the  terms  of  the 
same  unit  value  in  couplets,  and  make  the  statement  as  in  simple 
proportion  (767).     Then  find  the  required  term  by  (763,  3), 


132  PROPORTION. 

2.  If  20  horses  consume  36  tons  of  hay  in  9  mo.,  how 
many  tons  will  12  horses  consume  in  18  months  ? 

Drill  Exercise. — In  this  example,  what  is  the  denomination  of 
the  required  term  ?  What  given  number  has  the  same  unit  value  ? 
What  will  be  the  third  term  of  the  proportion?  The  fourth? 
How  many  couplets  are  there  ?  Give  them.  What  kind  of  a  ratio 
do  they  form  ?  How  is  the  antecedent  and  consequent  of  each 
couplet  determined  ?  How  is  a  compound  ratio  reduced  to  a  simple 
one  ?  Make  the  statement.  Is  the  required  term  a  mean  or  an 
extreme?    How  is  it  found?    (763,  3.) 

3.  If  $320  will  pay  the  board  of  4  persons  for  8  weeks, 
for  how  many  weeks  will  $800  pay  the  board  of  15 
persons  ? 

4.  If  a  man  walk  192  miles  in  6  days,  walking  8  hr.  a 
day,  how  far  can  he  walk  in  18  days,  walking  6  hr.  a  day  ? 

5.  If  6  laborers  can  dig  a  ditch  34  yards  long  in  10 
days,  how  many  days  wiB  20  laborers  require  to  dig  a 
ditch  170  yards  long? 

KuLE. — I.  Form  each  couplet  of  the  compound  ratio 
from  the  numbers  given^  iy  comparing  those  lohich  are  of 
the  same  unit  value,  arranging  the  terms  of  each  in  respect 
to  the  third  term  of  the  proportion,  as  if  it  were  the  first 
couplet  of  a  simple  proportion.     (767.) 

II.  Divide  the  product  of  the  second  and  third  terms  hy 
the  product  of  the  first  terms,  the  quotient  will  he  the  num- 
ber required. 

The  same  preparation  of  the  terms  by  reduction  is  to  be  observed 
as  in  simple  proportion. 

When  possible,  shorten  the  operation  by  cancellation.  When 
the  vertical  line  is  used,  write  the  factors  of  the  dividend  on  the 
right,  and  the  factors  of  the  divisor  with  x  on  the  left. 


PBOPORTION. 


133 


CAUSE     AND     EFFECT. 

773.  If  we  regard  the  conditions  of  each  problem  as 
the  comparison  of  two  causes  and  two  effects,  the  com- 
pound proportion  will  consist  of  two  ratios,  one  or  both 
of  which  may  be  compound,  and  the  required  term  will 
be  either  a  simple  cause,  or  effect,  or  a  single  element  of  a 
compound  cause,  or  effect. 


WRITTEN     EXEBC  IS  ES. 

774.  1.  If  8  men  earn  $320  in  8  days,  how  much  will 
12  men  earn  in  4  days  ? 


1st  cause. 

8  men  : 
8  days 


STATEMENT. 
2d.  cause. 

:  12  men  [  ^  ^ 
•    4  days  \ 

OPERATION. 

12x4x^^0^ 

^x^ 


Or, 


$320  :  fe    ^ 


=  $240 


12 

4 


$240 


Analysis.  — 
In  this  example 
the  first  cause  is 
8  men  at  work  8 
days,  the  second 
cause  is  13  men 
at  work  4  days  ; 
the  two  form  a 
compound  ratio. 
The  effect  of  the  first  cause  is  $320  earned,  the  effect  of  the  sec- 
ond cause  is  $x  earned,  and  is  the  required  term ;  the  two  effects 
form  a  simple  ratio. 

The  value  of  the  required  term  depends  upon  two  conditions : 
1st,  the  number  of  men  at  work  ;  2d,  the  number  of  days  they  work. 
Consider  each  condition  separately,  and  arrange  the  terms  of  the 
same  unit  value  in  couplets,  and  make  a  statement  in  the  same  man- 
ner as  in  simple  proportion.  The  required  term  being  an  extreme, 
is  found  by  (763,  3). 

2.  If  it  cost  $41.25  to  pave  a  sidewalk  5  ft.  wide  and 
75  ft.  long,  what  will  it  cost  to  pave  a  similar  walk  8  ft. 
wide  and  566  ft.  long  ? 


134  PROPORTION. 

3.  How  many  days  will  21  men  require  to  dig  a  ditch 
80  ft.  long,  3  ft.  wide,  and  8  ft.  deep,  if  7  men  can  dig  a 
ditch  60  ft.  long,  8  ft.  wide,  and  6  ft.  deep,  in  12  days  ? 

Or, 

80  X 

3  u  n 

8  ^m  ^(^8 


8 


OPERATION. 

8 
;?xlgx^0x$x^_8 
"^^     ^1x00x^x0    -3-^t^a- 

3  X—  2|  da. 

Analysis. — In  this  example  the  causes  and  the  effects  each  form 
a  compound  ratio.  The  required  term  is  an  element  of  the  second 
cause  and  a  mean.  Hence  divide  the  product  of  the  extremes  by 
the  product  of  the  given  means,  and  the  quotient  is  the  required 
factor  or  term,  3|  da.  (763,  2). 

4.  If  4  horses  consume  48  bushels  of  oats  in  12  days, 
how  many  bushels  will  20  horses  consume  in  8  weeks  ? 

EuLE. — I.  Of  the  given  numbers,  select  those  which  con- 
stitute the  causes,  and  those  which  constitute  the  effects, 
and  arrange  them  in  couplets  as  in  simple  cause  and  effect, 
putting  X  in  the  place  of  the  required  term. 

II.  If  the  required  term,  x,  he  an  extreme,  divide  the 
product  of  the  means  by  the  product  of  the  given  extremes  ; 
if  X  be  a  mean,  divide  the  product  of  the  extremes  by  the 
product  of  the  given  means ;  the  quotient  will  be  the  re- 
quired term. 

Solve  the  following  by  either  of  the  foregoing  methods  : 

5.  What  sum  of  money  will  produce  $300  in  8  mo.,  if 
$800  produce  $70  in  15  months  ? 


PROPOETION.  135 

6.  If  20  reams  of  paper  are  required  to  print  800  copies 
of  a  book  containing  230  pages  each,  40  lines  on  a  page, 
how  many  reams  are  required  to  print  3000  copies  of 
400  pages  each,  35  lines  on  a  page  ? 

7.  If  10  men  can  cut  46  cords  of  wood  in  18  da.,  work- 
ing 10  hr.  a  day,  how  many  cords  can  40  men  cut  in 
24  da.,  working  9  hr.  a  day  ? 

8.  What  is  the  cost  36^  yards  of  cloth  1|  yi  wide, 
if  2i  yards  If  yd.  wide,  cost  $3-37^  ? 

9.  A  contractor  employs  45  men  to  complete  a  work 
in  3  months ;  what  additional  number  of  men  must  he 
employ,  to  complete  the  work  in  2^  months? 

10.  If  a  vat  16  ft.  long,  7  ft  wide,  and  15  ft.  deep 
holds  384  barrels,  how  many  barrels  will  a  vat  17J  ft. 
long,  lOJ^  ft.  wide,  and  13  ft.  deep  hold  ? 

11.  What  is  the  weight  of  a  block  of  granite  8  ft.  long, 

4  ft.  wide,  and  10  in.  thick,  if  a  similar  block  10  ft.  long, 

5  ft.  wide,  and  16  in.  thick,  weigh  5200  pounds  ? 

12.  If  it  cost  $15  to  carry  20  tons  1^  miles,  what  will 
it  cost  to  carry  400  tons  ^  of  a  mile  ? 

13.  If  it  take  13500  bricks  to  build  a  wall  200  ft.  long, 
20  ft.  high,  and  16  in.  thick,  each  brick  being  8  in.  long, 

4  in.  wide,  and  2  in.  thick,  how  many  bricks  10  in.  long, 

5  in.  wide,  S^  in.  thick,  will  be  required  to  build  a  wall 
600  ft,  long,  24  ft.  high,  and  20  ft.  thick  ? 

14.  What  will  15  hogsheads  of  molasses  cost,  if  28^ 
gallons  cost  $7^  ? 

15.  At  6^d.  for  If  yards  of  cotton  cloth,  how  many 
yards  can  be  bought  for  £10  6s.  8d.  ? 

16.  If  $750  gain  $202.50  in  4  yr.  6  mo.^  what  sum  will 
gain  $155.52  in  1  yr.  6  mo.  ? 


136  PROPORTIOIf. 

17.  In  what  time  can  60  men  do  a  piece  of  work  that 
15  men  can  do  in  20  days  ? 

18.  If  2^  yd.  of  cloth  6  quarters  wide  can  be  made  from 
1  lb.  12  oz.  of  wool,  how  many  yards  of  cloth  4  quarters 
wide  can  be  made  from  70  lb.  of  wool  ? 

19.  If  the  use  of  $300  for  1  yr.  8  mo.  is  worth  $30,  how 
long,  at  the  same  rate,  may  $210.25  be  retained  to  be 
worth  $42,891  ? 

20.  A  farmer  has  hay  worth  $18  a  ton,  and  a  merchant 
has  flour  worth  $10  a  barrel.  If  the  farmer  ask  $21  for 
his  hay,  what  should  the  merchant  ask  for  his  flour  ? 

21.  How  many  men  will  be  required  to  dig  a  cellar 
45  ft.  long,  34.6  ft.  wide,  and  12.3  ft.  deep,  in  12  da.  of 
8.2  hr.  each,  if  6  men  can  dig  a  similar  one  22.5  ft.  long, 
17.3  ft.  wide,  and  10.25  ft.  deep,  in  3  da.  of  10.25  hr.  each? 

22.  If  a  bin  8  ft.  long,  4^  ft.  wide,  and  2^  ft.  deep, 
hold  67|^  bu.,  how  deep  must  another  bin  be  made,  that  is 
18  ft.  long  and  3|-  ft.  wide,  to  hold  450  bu.  ? 

23.  What  wiU  120  lb.  of  coffee  cost,  if  10  lb.  of  sugar 
cost  $1.25,  and  16  lb.  of  sugar  are  worth  5  lb.  of  coffee  ? 

24.  Two  men  haye  each  a  farm.  A's  farm  is  worth 
$48.75,  and  B's  $43| ;  but  in  trading  A  values  his  at  $60 
an  acre.     What  value  should  B  put  upon  his  ? 

25.  If  6  men  in  4  mo.,  working  26  da.  for  a  month, 
and  12  hr.  a  day,  can  set  the  type  for  24  books  of  300  pp. 
each,  60  lines  to  the  page,  12  words  to  the  line,  and  an 
average  of  6  letters  to  the  word,  in  how  many  months  of 
24  da.  each,  and  10  hr.  a  day,  can  8  men  and  4  boys  set 
tlie  type  for  10  books  of  240  pp.  each,  52  lines  to  the 
page,  16  words  to  the  line,  and  8  letters  to  the  word,  2 
boys  doing  as  much  as  1  man  ? 


ORAL    EXEMCISES. 

775.  1.  If  John  has  10  marbles,  William  15  marbles, 
and  Charles  25  marbles,  what  part  of  the  whole  has  each  ? 

2.  Two  men  bought  a  barrel  of  flour  for  $9,  the  first 
paying  $4  and  the  second  $5.  What  part  of  the  flour 
belongs  to  each  ? 

3.  Three  men  bought  108  sheep,  and  as  often  as  the 
first  paid  $3,  the  second  paid  $4,  and  the  third  $5.  How 
many  sheep  should  each  receive  ? 

4.  If  $45  be  divided  between  two  persons,  so  that  of 
every  $5,  one  receives  $2,  and  the  other  $3,  how  many 
dollars  does  each  receive  ? 

5.  Two  men  hired  a  pasture  for  $36 ;  one  put  in  2 
horses  for  3  weeks,  the  other  3  horses  for  4  weeks.  What 
should  each  pay  ? 

DEFINITIONS. 

776.  Partner  ship  is  the  association  of  two  or  more 
persons  under  a  certain  name,  for  the  transaction  of  busi- 
ness with  an  agreement  to  share  the  gains  and  losses. 

777.  A  Firnif  Company  or  Mouse  is  any  par- 
ticular partnership  association. 

778.  The  Capital  i&  the  money  or  property  invested 
by  the  partners,  called  also  Investment,  or  Joint- Stoch. 


138  PARTNERSHIP. 

imQ.  The  Resources  of  a  firm  are  the  amounts  due 
it,  together  with  the  property  of  all  kinds  belonging  to 
it ;  called  also  Assets,  or  Effects, 
HSO.  The  lAahilities  of  a  firm  are  its  debts, 
781.  The  Net  Capital  is  the  excess  of  resources 
over  liabilities. 

WJtXTTMN    EXERCISES, 

783.  To  apportion  grains  or  losses  according  to 
capital  invested. 

1.  A  and  B  engage  in  trade  ;  A  furnishes  $400  capital, 
B  1600.     They  gain  $250  ;  what  is  the  profit  of  each  ? 
1st  operation.     {By  Fractions.) 
$400,  A/s  investmeoit  =  y^^TT  =  i  of  ^^^  whole  capital. 
600,  B.'s  -  =  A'A  =  f     " 

$1000,  whole      " 

$250  X  f  rz:  $100,  A/s  share  of  the  gain. 
$250  X  f  =  $150,  B.'s     '* 

2d  operation.     {By  Proportion.) 
$1000  (whole  cap.)  :  $400  (A/s  in  v.)  : :  $250  (whole  gain) :  A/s  share. 
$1000  (whole  cap.)  :  $600  (B.'s  lav.)  : :  $250  (whole  gain) :  B/s  share. 

3d  operation.    {By  Percentage.) 

$250  gain  is  y%V^  :=:  25%  of  the  whole  capital. 

$400  X  .25  =  $100,  A/s  gain  ;    $600  x  .25  =  $150,  B.'s  gain. 

Analysis. — {1st  Method.)  Since  $400,  A.'s  investment,  is  -^-^y 
or  f ,  of  the  whole  capital,  he  is  entitled  to  f  of  the  gain,  or  $lt)0 ; 
and  B  is  entitled  to  f  of  the  gain,  or  $150. 

M  Method.  The  ratio  of  $1000,  the  whole  capital,  to  $400,  A.'s 
investment,  is  equal  to  the  ratio  of  $250,  the  whole  gain,  to  A.'s 
share  of  the  gain.     Hence  the  proportions,  etc. 

Sd  Method.  Since  the  gain  is  25  %  of  the  whole  capital,  each 
partner  is  entitled  to  25  %  of  his  investment  as  his  share  of  the  gain. 

The  third  method  (hy  dividend)  is  that  generally  adopted  by  joints 
stock  companies  having  numerous  shareholders. 


PARTNERSHIP.  139 

2.  At  the  end  of  the  year,  Norton,  Smith  &  Co.  take 
an  account  of  stock,  and  find  the  amount  of  merchandise, 
as  per  inventory,  to  be  $8400  ;  cash  on  hand,  $4850 ;  due 
from  sundry  persons,  $5273.  Their  debts  are  found  to 
amount  to  $4223.  S.  Norton's  investment  in  the  busi- 
ness is  $5000  ;  E.  Smith's,  $4000 ;  and  C.  Woodward's, 
$2000.  Make  a  statement  showing  the  resources,  lia- 
bihties,  net  capital,  and  net  gain ;  and  find  each  part- 
ner's share  of  the  gain. 

OPERATION. 

Mesources. 

Mdse.  as  per  inventory, $8400 

Cash  on  hand, 4850 

Debts  due  the  firm, •    5273 

$18523 

lA  abilities. 

Debts  due  to  sundry  persons, 4223 

Net  capital,      ....    $14300 

Investments, 

S.  Norton,  . $5000 

R.  Smith, 4000 

C.  Woodward, 2000 

Total  investments  .....    $11000 

Net  gain, $3300 

S.  Norton's  fractional  part,  ^\^^%  =  A  o^  $3^^^  =  $1^^0- 
R.  Smith's         ''  ''      Am  =  XT  of  $3300  =  $1300. 

C.  Woodward's  **  ''     A^  =  A  of  $3300  =  $  600. 

Proof.— $1500  +  $1200  +  $600  =  $3300,  total  gain. 


140  PAKTKERSHIP. 

Eul:e  1.  Find  what  fractional  part  each  partner^ s  in- 
vestment is  of  tJie  whole  capital^  and  take  such  part  of  the 
tvhole  gain  or  loss  for  his  share  of  the  gain  or  loss.     Or, 

2.  State  hy  proportion,  as  the  whole  capital  is  to  each 
partner's  investment t,  respectively,  so  is  the  whole  gain  or 
loss  to  each  partner's  share  of  the  gain  or  loss.     Or, 

3.  Find  what  per  cent,  the  gain  or  loss  is  of  the  whole 
capital,  and  taTce  that  per  cent,  of  each  partner's  invest- 
ment  for  his  share  of  the  gain  or  loss,  respectively, 

3.  A  furnishes  $4000,  B,  $2700,  and  C,  $2300,  to  pur- 
chase a  house,  which  they  rent  for  $720.  What  is  each 
one's  share  of  the  rent  ? 

4.  Four  persons  rent  a  farm  of  230  A.  64  P.  at  $7|^  an 
acre.  A  puts  in  288  sheep,  B,  320  sheep,  C,  384  sheep, 
and  D,  648  sheep  ;  what  rent  ought  each  to  pay  ? 

5.  Prime  &  Co.  fail  in  business ;  their  liabilities 
amount  to  $22000  ;  their  available  resources  to  $8800. 
They  owe  A  $4275,  and  B  $2175.50 :  what  will  each  of 
these  creditors  receive  ? 

6.  Four  persons  engage  in  manufacturing,  and  invest 
jointly  $22500.  At  the  expiration  of  a  certain  time,  A's 
share  of  the  gain  is  $2000,  B's  $2800.75,  C's  $1685.25, 
and  D's  $1014.     How  much  capital  did  each  put  in  ? 

7.  An  estate  worth  $10927.60  is  divided  between  two 
heirs  so  that  one  receives  ^  more  than  the  other.  What 
does  each  receive  ? 

8.  Three  persons  engage  in  the  lumber  trade  with  a 
joint  capital  of  $37680.  A  puts  in  $6  as  often  as  B  puts 
in  $10,  and  as  often  as  0  puts  in  $14.  Their  annual  gain 
is  equal  to  O's  stock.    What  is  each  partner's  gain  ? 


PARTKEESHIP.  141 

9.  Ames,  Lyon  &  Co.  close  business  in  the  following 
condition  :  notes  due  the  firm  to  the  amount  of  $24843.75, 
cash  in  hand,  $42375.80,  due  on  account,  $26500,  mer- 
chandise per  inventory,  $175840.  Notes  against  the 
firm,  $14058.75,  due  from  the  firm  on  account,  $12375.80. 
Ames  invested  $60000,  Lyon,  $40000,  and  Clark  $25000. 
Make  a  statement  showing  the  total  amount  of  resources, 
liabilities,  investments,  net  capital,  net  gain,  and  each 
partner's  share  of  the  gain. 

783.  To  apportion  ^ains  or  losses  according  to 
amount  of  capital  invested,  and  time  it  is  employed. 

1.  Three  partners.  A,  B,  and  C,  furnish  capital  as  fol- 
lows :  A,  $500  for  2  mo.;  B,  $400  for  3  mo.;  C,  $200  for 
4  mo.     They  gain  $600  ;  what  is  each  partner's  share  ? 

OPERATION. 

500  X  2  rr  1000  =  UU  =  i  X  )  (  $200,  A's  share. 

400  X  3  r=  1200  =  iUi  =  f  X  >-  $600  =  J  $240,  B's      " 
200  X  4  =_800  =  ^%%  =^\x  )  I  $160,  C's      " 

3000 

Analysis.— The  use  of  $500  for  2  mo.  is  the  same  as  the  use  of 
$1000  for  1  mo. ;  the  use  of  $400  for  3  mo.  is  the  same  as  that  of 
$1200  for  1  mo. ;  and  the  use  of  $200  for  4  mo.  is  the  same  as  that 
of  $800  for  1  mo.  Therefore  the  whole  capital  is  the  use  of  $3000 
for  1  mo.  ;  and  as  A*s  investment  is  $1000  for  1  mo.,  it  is  J  of  the 
capital,  and  hence  he  should  receive  J  of  the  gain,  or  $200.  For 
the  same  reason,  B  should  receive  f,  and  C  y%  of  the  gain,  or  $240 
and  $160,  respectively. 

The  other  methods  of  operation  may  be  applied  by  considering 
the  products  of  investment  and  time  as  shares  of  the  capital.  Thus, 
$600  is  20%  of  $3000;  and  20%  of  $1000,  $1200,  and  $800  will 
give  $200,  $240,  and  $160,  respectively,  the  shares  of  gain  required. 


142  PABTIS^EESHIP. 

2.  Barr,  Banks  &  Co.  gain  in  trade  $8000.  Barr  fur- 
nished $12000  for  6  mo..  Banks,  $10000  for  8  mo.,  and 
Butts  18000  for  11  mo.     Apportion  the  gain  ? 

EuLE  1. — Multiply  each  partners  capital  iy  the  time 
it  is  invested,  and  divide  the  whole  gain  or  loss  among  the 
partners  in  the  ratio  of  these  products.     Or, 

2.  State  hy  projjortion:  The  sum  of  the  products  is  to 
each  product,  as  the  whole  gain  or  loss  is  to  each  partner's 
gain  or  loss. 

3.  Jan.  ],  1876,  three  persons  began  business  with 
$1300  capital  furnished  by  A  ;  March  1,  B  put  in  $1000  ; 
Aug.  1,  C  put  in  $900.  The  profits  at  the  end  of  the 
year  were  $750.     Apportion  it. 

4.  In  a  partnership  for  2  years,  A  furnished  at  first 
$2000,  and  10  mo.  after  withdi^ew  $400  for  4  mo.,  and 
then  returned  it ;  B  at  first  put  in  $3000,  and  at  the  end 
of  4  mo.  1500  more,  but  drew  out  $1500  at  the  end  of  16 
mo.     The  whole  gain  was  $3372.  Find  the  share  of  each. 

5.  The  joint  capital  of  a  company  was  $5400,  which 
was  doubled  at  the  end  of  the  year.  A  put  in  ^  for  9  mo., 
B  f  for  6  mo.,  and  C  the  remainder  for  ]  year.  What  is 
each  one's  share  of  the  stock  at  the  end  of  the  year  ? 

6.  Crane,  Child  &  Coe,  forming  a  partnership  Jan.  1, 
1875,  invested  and  drew  out  as  follows:  Crane  invested 
$2000,  4  mo.  after  $1000  more,  and  at  the  end  of  9  mo. 
drew  out  $600.  Child  invested  $5000,  6  mo.  after  $1200 
more,  and  at  the  end  of  11  mo.  put  in  $2000  more.  Coe 
put  in  $6000,  4  mo.  after  drew  out  $4000,  and  at  the 
end  of  8  mo.  drew  out  $1000  more.  The  net  profits  for 
the  year  were  $7570.     Find  the  share  of  each. 


784.  Alligation  treats  of  mixing  or  compounding 
two  or  more  ingredients  of  different  values  or  qualities. 

785.  Alligation  Medial  is  the  process  of  finding 
the  mean  or  average  yalue  or  quality  of  several  ingredients. 

786.  Alligation  Alternate  is  the  process  of  find- 
ing the  proportional  quantities  to  be  used  in  any  required 
mixture. 


WRITTEN     EX  A  MPZJES, 

787.  1.  If  a  grocer  mix  8  lb.  of  tea  worth  $.60  a  pound 
with  6  lb.  at  $.70,  2  lb.  at  $1.10,  and  4  lb.  at  $1.20,  what 
is  1  lb.  of  the  mixture  worth  ? 

Analysis.  —Since  8  lb.  of  tea  at  $.60  is 
worth  $4.80,  and  6  lb.  at  $.70  is  worth 
$4.20,  and  2  lb.  at  $1.10  is  worth  $2.20, 
and  4  lb.  at  $1.20  is  worth  $4.80,  the  mix- 
ture of  20  lb.  is  worth  $16.  Hence  1  lb.  is 
worth  ^\  of  $16,  or  $16  -5-  20  =  $.80. 


2.  If  20  lb.  of  sugar  at  8  cents  be  mixed  with  24  lb.  at 
9  cents,  and  32  lb.  at  11  cents,  and  the  mixture  is  sold 
at  10  cents  a  pound,  what  is  the  gain  or  loss  on  the  whole  ? 

'RvL^.—Find  the  entire  cost  or  value  of  the  ingredients^ 
and  divide  it  iy  the  sum  of  the  simples. 


OPERATION. 

$.60 

X8  = 

$4.80 

.70 

X6  = 

4.20 

1.10 

X3  = 

2.20 

1.20 

x4  = 

4.80 

20  ) 

$16.00 

144  ,     ALLIGATION. 

3.  A  miller  mixes  18  bu.  of  wheat  at  $1.44  with  6  bu. 
at  $1.32,  6  bu.  at  $1.08,  and  12  bu.  at  $.84.  What  will  be 
his  gain  per  bushel  if  he  sell  the  mixture  at  $1.50  ? 

4.  Bought  24  cheeses,  each  weighing  25  lb.,  at  If  a 
pound ;  10,  weighing  40  lb.  each,  at  IW ;  and  4,  weigh- 
ing 50  lb.  each,  at  13^ ;  sold  the  whole  at  an  average 
price  of  ^^f  a  pound.     What  was  the  whole  gain  ? 

5.  A  drover  bought  84  sheep  at  $5  a  head ;  96  at  $4.75  ; 
and  130  at  $5^.  At  what  average  price  per  head  must 
he  sell  them  to  gain  20^  ? 

188.  To  find  the  proportional  parts  to  be  used, 
when  the  mean  price  of  a  mixture  and  the  prices  of 
the  simples  are  given. 

1.  What  relative  quantities  of  timothy  seed  worth  $2  a 
bushel,  and  clover  seed  worth  $7  a  bushel,  must  be  used 
to  form  a  mixture  worth  $5  a  bushel  ? 


i 


OPERATION.  ANAiiYSis. — Since  on  evAry  ingredient  usea 

2  )  whose  price  or  quality  is  less  than  the  mean 

•  Ans.  rate  there  will  be  a  gain,  and  on  every  ingre- 
dient whose  price  or  quality  is  greater  than 
the  mean  rate  there  will  be  a  loss^  and  sincp  the  gains  and  losses 
must  be  exactly  equal,  the  relative  quantities  used  of  each  should 
be  such  as  represent  the  unit  of  value.  By  selling  one  bushel  of 
timothy  seed  worth  $2,  for  $5,  there  is  a  gain  of  $3  ;  and  to  gain  $1 
would  require  i  of  a  bushel,  which  is  placed  opposite  the  2.  By 
selling  one  bushel  of  clover  seed  worth  $7,  for  $5,  there  is  a  loss 
of  $2  ;  and  to  lose  $1  would  require  J  of  a  bushel,  which  is  placed 
opposite  the  7. 

In  every  case,  to  find  the  unit  of  value,  divide  $1  by  the  gain  or 
loss  per  bushel  or  pound,  etc.  Hence,  if  every  time  J  of  a  bushel 
of  timothy  seed  is  taken,  J  of  a  bushel  of  clover  seed  is  taken,  the 
gain  and  loss  will  be  exactly  equal,  and  i  and  J  will  be  the  propor- 
tional quantities  required. 


ALLIGATION. 


145 


OPERATION. 


6 


f 

1 

% 

3 

4 

5 

3 

i 

4 

4 

4 

\ 

1 

1 

7 

1 

% 

2 

.10 

i 

3 

3 

To  express  the  proportional  numbers  in  integers,  reduce  these 
fractions  to  a  common  denominator,  and  use  their  numerators,  since 
fractions  having  a  common  denominator  are  to  each  other  as  their 
numerators  (241) ;  thus,  \  and  \  are  equal  to  f  and  |,  and  the  pro- 
portional quantities  are  2  bu.  of  timothy  seed  to  3  bu.  of  clover  seed. 

2.  What  proportions  of  teas  worth  respectively  3,  4,  7, 
and  10  shillings  a  pound,  must  be  taken  to  form  a  mix- 
ture worth  6  shillings  a  pound  ? 

Analysis. — To  preserve  the  equality 
of  gains  and  losses,  always  compare 
two  prices  or  simples,  one  greater  and 
one  less  than  the  mean  rate,  and  treat 
each  pair  or  couplet  as  a  separate  ex- 
ample. In  the  given  example  form  two 
couplets,  and  compare  either  3  and  10, 
4  and  7,  or  3  and  7,  4  and  10. 

We  find  that  J  of  a  lb.  at  3s.  must  be 
taken  to  qain  1  shilling,  and  \  of  a  lb. 
at  10s.  to  lose  1  shilling  ;  also  ^  of  a  lb.  at  4s.  to  gain  1  shilling,  and 
1  lb.  at  7s.  to  lose  1  shilling.  These  proportional  numbers,  obtained 
by  comparing  the  two  couplets,  are  placed  in  columns  1  and  2.  If, 
now,  the  numbers  in  columns  1  and  2  are  reduced  to  a  common  de- 
nominator, and  their  numerators  used,  the  integral  numbers  in 
columns  3  and  4  are  obtained,  which,  being  arranged  in  column  5, 
give  the  proportional  quantities  to  be  taken  of  each. 

It  will  be  seen  that  in  comparing  the  simples  of  any  couplet,  one 
of  which  is  greater,  and  the  other  less  than  the  mean  rate,  the  pro- 
portional number  finally  obtained  for  either  term  is  the  difference 
between  the  mean  rate  and  the  other  term.  Thus,  in  comparing  3 
and  10,  the  proportional  number  of  the  former  is  4,  which  is  the 
difference  between  10  and  the  mean  rate  6  ;  and  the  proportional 
number  of  the  latter  is  3,  which  is  the  difference  between  3  and  the 
mean  rate.  The  same  is  true  of  every  other  couplet.  Hence,  when 
the  simples  and  the  mean  rate  are  integers,  the  intermediate  steps 
taken  to  obtain  the  final  proportional  numbers  as  in  columns  1 ,  2,  3, 
and  4,  may  be  omitted,  and  the  same  results  readily  found  by  taking 
the  difference  between  each  simple  and  the  mean  rate,  and  placing 
it  opposite  the  one  with  which  it  is  compared. 


146  ALLIGATIOK. 

3.  In  what  proportions  must  sugars  worth  10  cents, 
11  cents,  and  14  cents  a  pound  be  used,  to  form  a  mix- 
ture worth  12  cents  a  pound  ? 

4.  A  farmer  has  sheep  worth  $4,  $5,  $6,  and  $8  per 
head.  What  number  may  he  sell  of  each  and  realize  an 
average  price  of  $5^  per  head  ? 

EuLE. — I.  Write  the  several  prices  or  qualities  in  a 
column^  and  the  mean  price  or  qicality  of  the  mixture  at 
the  left. 

II.  Forra  couplets  hy  comparing  any  price  or  quality 
less,  with  one  that  is  greater  than  the  mean  rate,  placing 
the  part  luhich  must  be  used  to  gain  1  of  the  mean  rate 
opposite  the  less  simple,  and  the  part  that  must  be  used  to 
lose  1  opposite  the  greater  simple,  and  do  the  same  for  each 
simple  in  every  couplet. 

III.  If  the  proportional  numbers  are  fractional,  they 
may  be  rediiced  to  integers,  and  if  two  or  more  stand  in 
the  same  horizontal  line,  they  must  be  added ;  the  final  re- 
sults will  be  the  proportional  quantities  required. 

1.  If  the  numbers  in  any  couplet  or  column  liave  a  common  fac- 
tor, it  may  be  rejected. 

2.  We  may  also  multiply  the  numbers  in  any  couplet  or  column 
by  any  multiplier  we  choose,  without  affecting  the  equality  of  the 
gains  and  losses,  and  thus  obtain  an  indefinite  number  of  results, 
any  one  of  which  being  taken  will  give  a  correct  final  result. 

5.  What  amount  of  flour  worth  $5^,  $6,  and  $?|  per 
barrel,  must  be  sold  to  realize  an  average  price  of  $6^  per 
barrel  ? 

6.  In  what  proportions  can  wine  worth  $1.20,  $1.80, 
and  $2.30  per  gallon  be  mixed  with  water  so  as  to  form  a 
mixture  worth  $1.50  per  gallon  ? 


ALLIGATION.  147 

789.  When  the  quantity  of  one  of  the  simples  is 
limited. 

1.  A  farmer  has  oats  worth  $.30,  com  worth  $.45,  and 
barley  worth  $.  84  a  bushel.  To  make  a  mixture  worth 
$.60  a  bushel,  and  which  shall  contain  48  bu.  of  corn, 
how  many  bushels  of  oats  and  barley  must  he  use  ? 

OPERATION.  Analysis.  —  By  tlie 

^30    JL  4  4    24^  same    process     as    in 

^^      .^  .  ^       ,^     .r.\     J  (788),  the  proportional 

60^5  ^  8       S    ^slAns.      quantities  of   each  are 

.84:    ^    ^    5    5     10     60  J  found  to  be   4  bu.  of 

oats,  8  of  corn,  and  10 
of  barley.  But  since  48  bu.  of  corn  is  6  times  the  proportional  num- 
ber 8,  to  preserve  the  equality  of  gain  and  loss  take  6  times  the 
proportional  quantity  of  each  of  the  other  simples,  or  6  x  4  =  24  bu. 
of  oats,  and  6  x  10  =:  60  bu.  of  barley.     Hence,  etc. 

2.  A  dairyman  bought  10  cows  at  $40  a  head.  How 
many  must  he  buy  at  $32,  $36,  and  $48  a  head,  so  that 
the  whole  may  average  $44  a  head  ? 

EuLE. — Find  the  proportional  quajitities  as  in  (788). 
Divide  the  given  quantity  hy  the  proportional  qua7itity  of 
the  same  ingredient,  and  multiply  each  of  the  other  propor- 
tional  quantities  iy  the  quotient  thus  ohtained, 

3.  A  grocer  having  teas  worth  $.80,  $1.20,  $1.50,  and 
$1.80  per  pound,  wishes  to  form  a  mixture  worth  $1.60  a 
pound,  and  use  20  lb.  of  that  worth  $1.50  a  pound. 

4.  Bought  12  yd.  of  cloth  for  $30.  How  many  yards 
must  I  buy  at  $3^  and  $1|  a  yard,  that  the  average  price 
of  the  whole  may  be  $2|  a  yard  ? 

5.  How  many  acres  of  land  worth  $70  an  acre  must  be 
added  to  a  farm  of  75  A.,  worth  $100  an  acre,  that  the 
average  value  may  be  $80  an  acre.'^ 


148 


ALLIGATION. 


OPEBATION 

r  6 

i 

3 

3 

37 

10  < 

7 
13 

i 

3 
3 

3 
3 

18 

37 

.13 

i 

4 

4 

36 

13 

108 

790.  When  the  quantity  of  the  whole  compound 
is  limited. 

1.  A  grocer  has  sugars  worth  6  cents,  7  cents,  12  cents, 
and  13  cents  per  pound.  He  wishes  to  make  a  mixture 
of  108  pounds,  worth  10  cents  a  pound ;  how  many 
pounds  of  each  kind  must  be  use  ? 

Analysis. — The  proportion- 
al quantities  of  each  simple 
found  by  (788)  are  3  lb.  at 
6  cts.,  2  lb.  at  7  cts.,  3  lb.  at  12 
cts.,  and  4  lb.  at  13  cts.  Add- 
ing the  proportional  quantities, 
the  mixture  is  but  12  lb , 
while  the  required  mixture  is 
108,  or  9  times  12.  If  the 
whole  mixture  is  to  be  9  times  as  much  as  the  sum  of  the  propor- 
tional quantities,  then  the  quantity  of  each  simple  used  must  be  9 
times  as  much  as  its  respective  proportional,  or  27  lb.  at  6  cts., 
18  lb.  at  7  cts.,  27  lb.  at  12  cts..  and  36  lb.  at  13  cts. 

2.  A  man  paid  $330  per  week  to  55  laborers,  consisting 
of  men,  women,  and  boys;  to  the  men  he  paid  $10  a 
week,  to  the  women  $2  a  week,  and  to  the  boys  $1  a  week ; 
how  many  were  there  of  each  ? 

EuLE. — Find  the  proportional  numbers  as  in  (788). 
Divide  the  given  quantity  ly  the  sum  of  the  proportional 
quantities^  and  multiply  each  of  the  proportional  quanti- 
ties ly  the  quotient  thus  obtained. 

3.  How  much  water  must  be  mixed  with  wine  worth 
$.90  a  gallon,  to  make  100  gal.  worth  $.60  a  gallon  ? 

4.  One  man  and  3  boys  received  $84  for  56  days'  labor ; 
the  man  received  $3  per  day,  and  the  boys  %\,  $|,  and 
$lf  respectively  ;  how  many  days  did  each  labor  ? 


REVIEW. 


149 


791. 


RATIO. 


PROPOR- 
TION. 


SYNOPSIS  FOR  EEVIEW. 

1.  Ratio.    2.  Sign  of  Ratio.    3.  Terms. 
4.  Antecedent.      5.   Consequent.      6. 

1.  Defs.  ^  Value  of  a  Ratio.  7.  Simple  Ratio. 
8.  Compound  Ratio.  9.  Reciprocal  of 
a  Ratio. 

2.  Formulas,  1,  2,  3. 

3.  General  Principles,  1,  2,  3. 

4.  General  Law. 


1.  Defs.  < 


''  1.  Proportion.      2.  Sign.      3.  Couplet. 
4  Proportional.    5.  Antecedents.    C. 
Consequents.    7.  Extremes.   8.  Means, 
9.  Mean  Proportional. 
2.  Principles,  1, 2,  3,  4. 


3.  Simple  Pro- 
portion. 


4.  Compound 
Proportion,  i 


.■U: 


...  Simple  Proportion, 
'     "   Statement, 

2.  Rule,  I,  II. 

3.  Cause  and  Effect, 
t  4.  Rule,  I,  II. 

"  1.  Def.    Compound  Proportion. 

2.  Rule,  I,  II. 

3.  Cause  and  Effect. 

4.  Rule,  I,  II. 


PARTNER. 
SHIP. 


ALLIGA- 
TION.       ^ 


{1.  Partnership.  2.  Firm,  Company,  or 
House.  3.  Capital.  4.  Resources. 
5.  Liabilities.      6.  Net  Capital. 

2.  782.    Rule,  1,  2,  3. 

3.  783.    Rule,  1,  2. 


1.  Defs. 


■]'■ 


Alligation.      2.    Alligation  MediaL 
3.  Alligation  Alternate. 

2.  787.    Rule. 

3.  788.    Rule,  I,  II,  III. 

4.  789.    Rule. 

5.  790.    Rule. 


150  GENERAL     REVIEW. 

TEST    PROBLEMS. 

792.  1.  The  sum  of  two  numbers  is  120,  and  their  dif- 
ference is  equal  to  ^  of  the  greater.     Find  the  numbers. 

2.  E's  age  is  H  times  the  age  of  D,  and  F's  age  is  2^ 
times  the  age  of  both,  and  the  sum  of  their  ages  is  124. 
What  is  the  age  of  each  ? 

3.  If  7  bu.  of  wheat  are  worth  10  bu.  of  rye,  and  5  bu. 
of  rye  are  worth  14  bu.  of  oats,  and  6  bu.  of  oats  are 
worth  $6,  how  many  bushels  of  wheat  will  $60  buy? 

4.  A  mechanic  was  engaged  to  labor  20  days,  on  the 
conditions  that  he  was  to  receive  15  a  day  for  every  day 
he  worked,  and  to  forfeit  $2  a  day  for  every  day  he  was 
idle  ;  at  the  end  of  the  time  he  received  $86.  How  many 
days  did  he  labor? 

5.  One  man  can  build  a  fence  in  18  da.,  working  10  lir. 
a  day ;  another  can  build  it  in  9  da.,  working  8  hr.  a  day. 
In  how  many  days  can  both  together  build  it,  if  they 
work  6  hours  a  day  ? 

6.  If  6  boxes  of  starch  and  7  boxes  of  soap  cost  $33, 
and  12  boxes  of  starch  and  10  boxes  of  soap  cost  $54,  what 
is  the  price  of  1  box  of  each? 

7.  Three  men  agree  to  build  a  bam  for  $540.  The  first 
and  second  can  do  the  work  in  16  da.,  the  second  and 
third  in  13 J^  da.,  and  the  first  and  third  in  llf  da.  In 
how  many  days  can  all  do  it  working  together  ?  In  how 
many  days  can  each  do  it  alone  ?  What  part  of  the  pay 
should  each  receive  ? 

8.  A  dealer  paid  $182  for  20  barrels  of  flour,  giving  $10 
for  first  quality,  and  $7  for  second  quality.  How  many 
barrels  were  there  of  each  ? 


GENERAL     REVIEW.  151 

9.  The  hour  and  minute  hands  of  a  clock  are  together 
at  12  M.  When  will  they  be  exactly  together  the  third 
time  after  this  ? 

10.  Bought  15  bu.  of  wheat  and  30  bu.  of  oats  for  $35, 
and  9  bu.  of  wheat  and  6  bu.  of  oats  for  $15.  What  was 
the  price  per  bushel  of  each  ? 

11.  If  Ames  can  do  as  much  work  in  3  days  as  Jones 
can  do  in  4^  days,  and  Jones  can  do  as  much  in  9  days  as 
Smith  can  do  in  12  days,  and  Smith  as  much  in  10  days 
as  Eay  in  8  days,  how  many  days'  work  done  by  Kay  are 
equal  to  5  days  done  by  Ames  ? 

12.  A  merchant  bought  40  pieces  of  cloth,  each  piece 
containing  25  yd. ,  at  $4f  per  yard,  on  9  mo.  credit,  and 
sold  the  same  at  $4f  per  yard,  on  4  mo.  credit.  Find  his 
net  cash  gain,  money  being  worth  6%. 

13.  There  are  70  bu.  of  grain  in  2  bins,  and  in  1  bin 
are  10  bu.  less  than  f  as  much  as  there  is  in  the  other. 
How  many  bushels  in  the  larger  bin  ? 

,  14.  Three  men  can  perform  a  piece  of  work  in  12  hr. ; 
A  and  B  can  do  it  in  16  hr.,  A  and  C  in  18  hr.  What 
part  of  the  work  can  B  and  C  do  in  9\  hours  ? 

15.  What  per  cent,  in  advance  of  the  cost  must  a  mer- 
chant mark  his  goods,  so  that  after  allowing  6%  of  his 
sales  for  bad  debts,  an  average  credit  of  6  mo.,  and  4:%  of 
the  cost  of  the  goods  for  his  expenses,  he  may  make  a 
clear  gain  of  12|^^  on  the  first  cost  of  the  goods,  money 
being  worth  7%  ? 

16.  An  elder  brother's  fortune  is  1^  times  his  younger 
brother's  ;  the  interest  of  J  of  the  elder  brother's  fortune 
and  -J-  of  the  younger's  for  5  years,  at  6^,  is  $2400.  What 
is  the  fortune  of  each  ? 


152  GENERAL     REVIEW. 

17.  The  top  of  Trinity  Church  steeple  in  New  York  is 
368  ft.  from  the  ground;  f  the  height  of  the  steeple 
above  the  church  plus  12  ft.  is  equal  to  the  height  of  the 
church.    Find  the  height  of  the  steeple  above  the  church  ? 

18.  Two  persons  have  the  same  income  :  A  saves  J  of 
his,  but  B  by  spending  $300  a  year  more  than  A,  at  the 
end  of  2  years  is  $200  in  debt.     What  is  their  income  ? 

19.  Divide  $2520  among  3  persons,  so  that  the  second 
shall  have  f  as  much  as  the  first,  and  the  third  ^  as  much 
as  the  other  two.     What  is  the  share  of  each  ? 

20.  A  man  owes  a  debt  to  be  paid  in  4  equal  install- 
ments of  4,  9,  12,  and  20  months  respectively ;  a  discount 
of  6%  being  allowed,  he  finds  that  $1500  ready  money  will 
pay  the  debt.    What  is  the  amount  of  the  debt  ? 

21.  A  quantity  of  flour  is  to  be  distributed  among  some 
poor  families ;  if  50  lb.  are  given  to  each  family,  there 
will  be  6  lb.  left ;  if  51  lb.  are  given  to  each,  there  will  be 
wanting  4  lb.     What  is  the  quantity  of  flour  ? 

22.  1  have  three  notes  payable  as  follows  :  one  for  $400, 
due  Jan.  1,  1875  ;  another  for  $700,  due  Sept.  1,  1875 ; 
and  another  for  $1000,  due  April  1,  1876.  What  is  the 
average  of  maturity  ? 

23.  An  estate  worth  $123251.82  is  left  to  four  sons, 
whose  ages  are  19,  17,  13,  and  11  years,  respectively,  and 
is  to  be  so  divided  that  each  part  being  put  out  at  7% 
simple  interest,  the  amounts  shall  be  equal  when  they 
become  21  years  of  age.     What  are  the  parts  ? 

24.  If  a  piece  of  silk  cost  $1 .20  a  yard,  at  what  price 
must  it  be  marked  that  it  may  be  sold  at  10^  less  than 
the  marked  price,  and  still  make  a  profit  of  20^  ? 

25.  A  farmer  sold  100  geese  and  turkeys  ;  lor  the  geese 


GENERAL     EEVIEW.  153 

he  received  $.75  apiece,  and  for  the  turkeys  $1.25  apiece, 
and  for  the  whole  $104.     What  was  the  number  of  each  ? 

26.  A  man  left  his  property  to  three  sons  ;  to  A  ^  want- 
ing $180,  to  B  i,  and  to  C  the  rest,  which  was  $590  less 
than  A  and  B  received.     What  was  the  whole  estate  ? 

27.  What  is  the  simple  interest  and  the  amount ;  the 
compound  interest  and  amount ;  the  present  worth  and 
the  true  discount ;  the  bank  discount  and  the  proceeds 
of  $1920,  for  2  yr.  5  mo.  12  da.,  at  6%  ?  Also  the  face  of 
the  note,  which  when  discounted  at  a  bank  for  the  same 
time,  and  at  the  same  rate,  will  produce  the  same  sum  ? 

28.  Divide  $1500  among  3  persons,  so  that  the  share 
of  the  second  may  be  ^  greater  than  that  of  the  first,  and 
the  share  of  the  third  ^  greater  than  that  of  the  second. 

29.  A  merchant  owes  for  three  bills  of  goods  as  follows : 
$500  due  March  1,  $800  due  June  1,  and  $600  due  Aug.  1. 
He  wishes  to  give  two  notes  for  the  amount,  one  for  $1000, 
payable  April  1 ;  what  must  be  the  face,  and  when  the 
maturity,  of  the  other  ? 

30.  A  man  in  New  York  purchased  a  draft  on  Chicago 
for  $10640,  drawn  at  60  da.,  $10302.18.  What  was  the 
course  of  exchange  ? 

31.  B.  B.  Northrop,  through  his  broker,  invested  a 
certain  sum  in  U.  S.  6's,  5-20,  at  107|^,  and  twice  as  much 
in  U.  S.  5's  of  '81,  at  98|,  brokerage  on  each,  ^%.  His 
income  from  both  investments  is  $1674.  How  much  did 
he  invest  in  each  kind  of  stock  ? 

32.  A,  B,  and  C  are  under  a  joint  contract  to  furnish 
6000  bu.  of  corn,  at  $.48  a  bushel ;  A's  corn  is  worth  $.45, 
B's  $.51,  and  C's  $.54  ;  how  many  bushels  must  each  put 
into  the  mixture  that  the  contract  mav  be  fulfilled  ? 


154  GEKERAL     REVIEW. 

33.  A  cask  contains  42J  U.  S.  gallons  of  wine,  worth 
$4|-  per  gallon.  How  much  less  will  it  cost  in  U.  S. 
money,  at  the  rate  of  £1  2s.  per  the  Imperial  gallon  ? 

34.  A  garden  400  ft.  long  and  300  ft.  wide  has  a  walk 
20  ft.  wide  laid  off  from  each  of  its  two  sides.  What  is 
the  ratio  between  the  area  of  the  walk  and  the  area  of 
what  remains  ? 

35.  A  commission  merchant  in  Charleston  received  into 
his  store  on  May  1,  1875,  1000  bbl.  of  flour,  paying  as 
charges  on  the  same  day,  freight  $175.48,  cartage  $56.25, 
and  cooperage  $8.37.  He  sold  out  the  shipment  as  fol- 
lows: June  3,  200  bbl.  @  $6.25  ;  June  30,  350  bbl.  @ 
$6.50  :  July  29,  400  bbl.  @  $6.12|^ ;  Aug.  6,  50  bbl.  @  $6. 
Eequired,  the  net  proceeds,  and  the  date  when  they  shall 
be  accredited  to  the  owner,  allowing  commission  at  3^%, 
and  storage  at  2  cents  per  week  per  bbl. 

"^  36.  Three  men  engage  in  manufacturing.  L  puts  in 
$3840  for  6  mo.  ;  M,  a  sum  not  specified  for  12  mo. ;  and 
N,  $2560  for  a  time  not  specified.  L  received  $4800  for 
his  capital  and  profits  ;  M,  $9600  for  his  ;  and  N,  $4160 
for  his.     Eequired,  M's  capital  and  N's  time. 

37.  My  expenditures  in  building  a  house,  in  the  year 
1874,  were  as  follows  :  Jan.  16,  $536.78  ;  Feb.  20,  $425.36  ; 
March  4,  $259.25  ;  April  24,  $786.36.  At  the  last  date  I 
sold  the  house  for  exactly  what  it  cost,  interest  at  6  per 
cent,  on  the  money  expended  added,  and  took  the 
purchaser's  note  for  the  amount.  What  was  the  face  of 
the  note  ? 

38.  A  man  bought  a  farm  for  $6000,  and  agreed  to  pay 
principal  and  interest  in  3  equal  annual  installments. 
What  was  the  annual  payment,  interest  being  6%  ? 


ORAL      BXEMCISES . 

793.  1.  What  is  the  product  of  3  used  twice  as  a 
factor  ? 

2.  What  is  the  product  of  3  used  3  times  as  factor  . 

3.  What  is  the  product  of  4  used  3  times  as  a  factor  ? 

4.  What  is  the  result  of  using  5  twice  as  a  factor  ? 

5.  What  is  the  product  of  -^  used  twice  as  a  factor  ? 

6.  What  is  the  result  of  using  f  twice  as  a  factor? 
f ,  three  times  as  a  factor  ? 

7.  What  number  will  be  produced  by  using  .3  twice  as 
a  factor?    .7,  twice  ?    A,  three  times  ?    .05,  twice  ? 

8.  A  room  is  9  ft.  on  each  side  ;  how  many  square  feet 
in  the  floor  ? 

9.  A  cubical  block  of  stone  is  4  ft.  on  each  edge  ;  how 
many  cubic  feet  does  it  contain  ? 

DEFINITIONS. 


794.  A  Power  of  a  number  is  the  product  of  factors, 
each  of  which  is  equal  to  that  number.  Thus,  27  is  the 
third  power  of  3,  since  27  =  3x3x3. 

795.  Involution  is  the  process  of  finding  any  power 
of  a  number. 


156 


INVOLUTION. 


796.  The  JBase  or  Root  of  a  power  is  one  of  the 
equal  factors  of  the  power.  Thus,  27  is  the  third  power 
of  3,  and  3  is  the  base,  or  root,  of  that  power. 

797.  The  ExiJOiient  of  a  power  is  a  number  placed 
at  the  right  of  the  base  and  a  little  above  it,  to  show  how 
many  times  it  is  used  as  a  factor  to  produce  the  power. 
It  also  denotes  the  degree  of  the  power.     Thus, 

3^  or  3  =    3,  the^r^^      power  of  3. 

32  z=  3  X  3  =9,  the  second  power  of  3. 

33  zzz  3  X  3  X  3  =  27,  the  third    power  of  3. 

34  1=  3  X  3  X  3  x3  =  81,  the/o^^r^Apowerof3. 


3-^  =  3x3  =  9 


^ 


^^I 


\      -^    -^^5^ 

^ 

iii; 

,:',.;r|!'ll 

III 

'itll'iiiSi 

ililHili^illMili 

ii!|iV"'.;    " 
miiiiyiiiumimi 

8»  =  3x8x3  =  27 


798.  The  Square  of  a  num- 
ber is  its  second  power,  so  called 
because  the  number  of  superficial 
units  in  a  square  is  equal  to  the 
second  power  of  the  number  of 
linear  units  in  one  of  its  sides. 


799.  The  O^ft^  of  a  num- 
ber is  its  third  power,  so 
called  because  the  number  of 
units  of  Yolume  in  a  cube  is 
equal  to  the  third  power  of 
the  number  of  linear  units 
in  one  of  its  edges. 


800.  A  Perfect  Power  is  a  number  which  can  be 
resolved  into  equal  factors.  Thus,  25  is  a  perfect  power 
of  the  second  degree,  and  27  is  a  perfect  power  of  the 
third  degree. 


IN^VOLUTION.  157 

801,  Pkin^ciple. — The  sum  of  the  exponents  of  two 
powers  of  the  same  number  is  equal  to  the  exponent  of  the 
product  of  those  powers.  Thus,  2^  x  2^=25 ;  for  22=:2  x  2, 
and  2^=2  x  2  x  2  ;  hence  2^  x  23=2  x  2  x  2  x  2  x  2=25. 

WRITTEN     JEXEHCISES. 

803.  To  find  any  power  of  a  number. 

1.  Find  the  third  power  of  35. 

OPERATION.  Anai^tsis. — Since  using 

35  =  351 ;    35  X  35  =  35^  =z  1225     any  number  three  times 

1225  X  35  =  353  =  42875  Z  \  ^^^"^^  ^jl^T'  *^^ 

third  power  of  that  num- 
ber (797),  35  X  35  X  35  :rr  353  =  42875. 

2.  Find  the  square  of  37.     Of  42.'    Of  56.    Of  75. 

3.  Find  the  cube  of  15.     Of  18.     Of  42.     Of  54. 

4.  What  is  the  value  of  63^  ?  of  48^  ?  of  32^  ?  of  12^  ? 

EuLE. — Find  the  product  of  as  many  factors^  each 
equal  to  the  given  number y  as  there  are  units  in  the  expo- 
nent of  the  required  power. 

5.  What  is  the  third  power  of  f  ? 

r^  /^x2      .     ^     d       4x4x4      43       64 

Operation.— (If  =  f  x  f  x  |  =  ^—^—z  =  -^  =  :r~, 
^^        ^     *     *      5x5x5      5^      125 

EuLE. — A  fraction  may  be  raised  to  any  power  by  in* 

volving  each  of  its  terms  separately  to  the  required  power. 

6.  What  is  the  square  of  ^^^  ?    The  cube  of  |f  ? 

7.  Eaise  ^  to  the  4th  power.    2|^  to  the  5th  power. 
Find  the  required  power  of  the  following  : 


8. 

35.42. 

13. 

.03438. 

16. 

(182i)2. 

9. 

1063. 

13. 

.5«. 

17. 

(4.07i)2, 

10. 

(44i)2. 

14. 

36.03». 

18. 

(1t^)=- 

11. 

(H)*. 

15. 

.403163 

19. 

.00638. 

158. 


INVOLUTION. 


Find  the  value  of  each  of  the  following  expressions  ; 


20.  4.63  X  253. 

21.  6.754 -(7^)2. 

22.  -Jof(i)3x(3|)2. 
26.  (43x56x123)-^ 


23.  8«  -f-  .4096. 

24.  2.53x(12|)2. 

25.  (7.5)3  _^  (1^)3. 
(42x104x32). 


FORMATION  OF  SQUARES  AND  CUBES  BY  THE  ANALTT 
ICAL  METHOD. 

803.  To  find  the  square  of  a  number  in  terms  of 
its  tens  and  units. 

1.  Find  the  square  of  27  in  terms  of  its  tens  and  units. 

OPERATION. 


27  = 
27  = 


20  +  7 
20  +  7 


189=  (20x7)+ 72 

540=        202 +(20x7) 
729  =  202+(2x20x7)+72 


Analysis. — The  product  of  20 
+  7  by  7  is  20  X  7  +  7\  and  the 
product  of  20  +  7  by  20  is  20^  +  (20 
x7);  hence  202  +  (2x20x7)  +  72, 
which  is  the  sum  of  these  partial 
products,  is  the  square  of  20  +  7 
or  27. 


Principle. — The  square  of  a  number  consisting  of  tens 
and  units  is  equal  to  the  sum  of  the  squares  of  the  tens 
and  units  increased  iy  twice  their  product. 

Geometrical  Illustration. 

Let  ABCD  be  a  square,  each  side 
of  which  is  27  feet,  and  let  lines  be 
drawn  as  represented  in  the  figure.  It 
is  evident  that  the  square  ABCD  (27^) 
is  equal  to  the  sum  of  two  squares,  one 
of  which  is  the  square  of  tens  (20^),  the 
other  the  square  of  the  units  (7^),  to- 
gether with  two  rectangles  each  of 
whose  areas  is  20  x  7. 


INVOLUTION.  159 

2.  What  is  the  square  of  37  ? 

2  X  30  X  7  =    420 

7'=r      49 

372  =  1369     (803,  Prin.) 

3.  Find  the  square  of  42  in  terms  of  its  tens  and  units. 
In  like  manner  find  the  square 

4.  Of  48. 


5.  Of  56. 


6.  Of    98. 

7.  Of  125. 


8.  Of  105. 

9.  Of  225. 


10.  Of  197. 

11.  Of  342. 


804.  To  find  the  cube  of  a  number  in  terms  of 
its  tens  and  units. 

1.  Find  the  cube  of  25  in  terms  of  its  tens  and  units. 

OPERATION. 

252=  202+ (2x20x5) +  58 

25  =    20  +  5 

252x   5  =  (202x5)  +  (2x20x52)  +  53 

25^x20  zir203  + (2x20^x5)+       (20x5^) 

253  =  203+{3  X 202 x5)  +  (3x 20x52) +  53 

Analysis.  —The  square  of  35  is  20^  4-  (2  x  20  x  5)  +  5^.  (803,  Prin.) 
Multiplying  this  by  20  -»-  5  gives  the  cvbe  of  25. 

2.  Find  the  cube  of  34  in  terms  of  its  tens  and  units. 

Principle. — The  cube  of  a  number  consisting  of  tens 
and  units  is  equal  to  the  cube  of  the  tens,  plus  three  times 
the  product  of  the  square  of  the  tens  by  the  units,  plus 
three  times  the  product  of  the  tens  by  the  square  of  the 
units,  plus  the  cube  of  the  units. 


160 


INVOLUTIOIT. 


Geometrical  Illustration. 


Fig.  1. 


The  volume  of  the 
cube  marked  A,  Fig.  1 , 
is  20^  ;  the  volume  of 
each  of  the  rectangu- 
lar solids  marked  B  is 
20  X  20  X  5,  or  20-^  x  5  ; 
the  volume  of  each  of 
the  rectangular  solids 
marked  C,  in  Fig.  2,  is 

B  20  X  5  X  5,    or   20  x  5^  ; 

|B  and  the  volume  of  the 
small  cube  marked  D 
is  5^.  It  is  evident, 
that  if  all  these  solids 
are  put  together  ais 
represented  in  Fig.  3, 
a  cube  will  be  formed, 
each  edge  of  which 
is  25. 

3.  Find  the  cube 
of  46? 

OPERATION. 

403=  64000 

402x6x3  =  28800 

40x62x3=   4320 

63=     216 

463=97336 

In  like  manner 
find  the  cube 

4.  Of  48. 

5.  Of  64. 

6.  Of  95. 

7.  Of  125. 


805.  1.  What  are  the  two  equal  factors  of  25  ?    36  ? 

2.  What  are  the  three  equal  factors  of  27  ?    64  ?    125  ? 

3.  What  are  the  four  equal  factors  of  16  ?    81  ?    256  ? 

4.  Of  what  is  81  the  2d  power  ?    The  4th  power  ? 

DBFUnTITIONS. 

806.  The  Square  Hoot  of  a  number  is  one  of  the 
two  equal  factors  of  that  number ;  the  Cube  Root  is 

one  of  the  three  equal  factors  of  that  number,  etc. 
Thus,  3  is  the  square  root  of  9,  2  is  the  cube  root  of  8,  etc. 

807.  Evolution  is  the  process  of  finding  the  root 
of  any  power  of  a  number. 

808.  The  Madical  Sign  is  V.  When  prefixed  to 
a  number,  it  indicates  that  some  root  of  it  is  to  be  found. 

809.  The  Index  of  the  root  is  a  small  figure  placed 
aboye  the  radical  sign  to  denote  what  root  is  to  be  found. 
When  no  index  is  written,  the  index  2  is  understood. 

Thus,  /\/T00  denotes  the  square  root  of  100 ;  \/\2^  denotes  the 
cube  root  of  125 ;  v^256  denotes  the  fourtJi  root  of  256  ;  and  so  on, 

Evolution,  or  both  involution  and  evolution,  may  be  indicated  in 
the  same  expression  by  a  fractional  exponent,  the  numerator  de- 
noting the  required  power  of  the  given  number,  and  the  denomina- 
tor the  root  of  that  power  of  the  number.     Thus, 

Oi  is  equivalent  to  y^O  ;  643 ,  to  /^64  ;  and  8f ,  to  the  cube  root 
of  the  second  power  of  8,  equivalent  to  >y/8^,  etc. 


163  EVOLUTION". 

EVOLUTION    BY    FACTORING. 

WJRITTJSN    EXERCISES, 

810.  To  find  any  root  of  a  number  by  factoring. 

1.  Find  the  cube  root  of  1728. 

OPERATION. 

3)1728 

Q  \  K  7  « 

-^  Analysis. — A  number  that  is  a  perfect  cube,  is 

3)192  composed  of  three  equal  factors,  and  one  of  them 

TTTT  is  the  cube  root  of  that  number. 
^y^  The  prime  factors  of  1728  are  3,  3,  3,  2,  2,  2, 

2)3  2  2,  2,  2  ;  hence  1728  =  (3  x  2  x  2)  x  (3  x  2  x  2)  x 

aTTfi  (3x2x2);  therefore  the  cube  root  of  1728  is 

<-—  (3  X  2  X  2),  or  12. 

2)8 

2)4 

2 

EuLE. — Resolve  the  given  number  into  its  prime  factors  ; 
then,  to  produce  the  square  root,  take  one  of  every  two  equal 
factors ;  to  produce  the  cube  root  take  one  of  every  three 
equal  factors  ;  and  so  on, 

2.  Find  the  square  root  of  64.  Of  256.  Of  576.  Of  6561. 

3.  Find  the  cube  root  of  729.  Of  2744.  Of  9261.  Of  3375. 

GENERAL    METHOD    OF    SQUARE    ROOT. 

811.  A  Perfect  Square  is  a  number  which  has 
an  exact  square  root. 

813.  Pkinciples. — 1.  The  square  of  a  number  ex- 
pressed by  a  single  figure  contains  no  figure  of  a  higher 
order  than  tens, 

2.  The  square  of  tens  contains  no  significant  figure  of  a 
lower  order  than  hundreds,  nor  of  a  higher  order  than 
thousands. 


SQUARE     ROOT.  163 

3.  The  square  of  a  number  contains  twice  as  many 
figures  as  the  number,  or  twice  as  many  less  one.     Thus, 

12  =         1,  102  =:  100, 

92  =       81,  1002  =       10000, 

992  =  9801,  10002  z=  1000000. 

Hence, 

4.  If  any  perfect  square  be  separated  into  periods  of  two 
figiores  each,  beginning  with  units^  place,  the  number  of 
periods  will  be  equal  to  the  number  of  figures  in  the  square 
root  of  that  number. 

If  the  number  of  figures  in  the  number  is  odd^  the  left-hand 
period  will  contain  only  one  figure. 

WniTTEN    X:XEItCISE8. 

813.  To  find  the  square  root  of  a  number. 

1.  Find  the  square  root  of  4356. 

OPERATION.  Analysis.— Since  4356  con- 

43,56(60  +  6  =  66     si^ts    of    two    periods,    its 

gQ2 3600  square  root  will  consist  of 

two  figures  (812,  Prin.  4). 

130  +  6  =  126  )  756  Since  56  cannot  be  a  part  of 

756  tlie  square  of  the  tens  (812, 

Prin.  2),  the  tens  of  the  root 
must  be  found  from  the  first  period  43. 

The  greatest  number  of  tens  whose  square  is  contained  in  4300 
is  6.  Subtracting  3600,  which  is  the  square  of  6  tens,  from  the 
given  number,  the  remainder  is  756.  This  remainder  is  composed 
of  twice  the  product  of  the  tens  by  the  units,  and  the  square  of  the 
units  (803,  Prin.).  But  the  product  of  tens  by  units  cannot 
be  of  a  lower  order  than  tens  ;  hence  the  last  figure  6  cannot  be  a 
part  of  twice  the  product  of  the  tens  by  the  units ;  this  double 
product  must  therefore  be  found  in  the  part  750. 

Now,  if  we  double  the  tens  of  the  root  and  divide  750  by  the 
result,  the  quotient  6    will  be  the  units'  figure  of  the  root,  or  a 


164  EVOLUTION. 

figure  greater  thaD  the  units'  figure.  This  quotient  cannot  be  too 
small,  for  the  part  750  is  at  least  equal  to  twice  the  product  of  the 
tens  by  the  units  ;  but  it  may  be  too  large,  for  the  part  750,  be- 
sides the  double  product  of  the  tens  by  the  units,  may  contain  tens 
arising  from  the  square  of  the  units.  (812,  Prin.  1.)  Subtracting 
6  X  120  + 6*  or  6  x  120  +  6  from  756,  nothing  remains.  Hence  66  is 
the  required  root. 

1.  In  this  example,  120  is  a  partial  or  trial  divisor,  and  126  is  a 
complete  divisor. 

2.  If  the  root  contains  more  than  two  figures,  it  may  be  found  by 
a  similar  process,  as  in  the  following  example,  where  it  will  be 
seen  that  the  partial  divisor  at  each  step  is  obtained  by  doubling 
that  part  of  the  root  already  found. 

2.  Find  the  square  root  of  186634. 

OPERATION. 

18,66,24(400  +  30  +  2=432 

16    00   00  nru  '    X.  *!,  .    W 

The  ciphers  on  the  right 

400  X  2  +  30  =  830  )  2  Q^  24      are  usually  omitted  for  the 
2  49  00      sake  of  brevity.    Thus, 

400x2  +  30x2  +  2=862)1724  18,66,24(432 

1724  16 

83)266,  etc. 

3.  Find  the  square  root  of  7225. 

4.  What  is  the  square  root  of  58564. 

Rule. — I.  Separate  the  given  number  into  periods  of  two 
figures  each,  beginning  at  the  units'  place. 

II.  Find  the  greatest  number  whose  square  is  contained 
in  the  period  on  the  left ;  this  will  be  the  first  figure  in  the 
root.  Subtract  the  square  of  this  figure  from  the  period  on 
the  left,  and  to  the  remainder  annex  the  next  period  to  form 
a  dividend. 


SQUARE     ROOT. 


165 


III.  Divide  this  dividend,  omitting  the  figure  on  the 
right,  iy  double  the  part  of  the  root  already  found,  and 
annex  the  quotient  to  that  part,  and  also  to  the  divisor ; 
then  multiply  the  divisor  thus  completed  ly  the  figure  of 
the  root  last  obtained,  and  subtract  the  product  from  the 
dividend. 

IV.  //  there  are  more  periods  to  be  brought  down,  con- 
tinue the  operation  in  the  same  manner  as  before. 

1.  If  a  cipher  occur  in  the  root,  annex  a  cipher  to  the  trial  divi- 
sor, and  another  period  to  the  dividend,  and  proceed  as  before. 

2.  If  there  is  a  remainder  after  the  root  of  the  last  period  is 
found,  annex  periods  of  ciphers  and  continue  the  root  to  as  many 
decimal  places  as  are  required. 

Find  the  square  root 


5.  Of    9G04. 

7.  Of    11881. 

9.  Of    2050624. 

6.   Of  13225. 

8.  Of  994009. 

10.  Of  29855296. 

11.  Find  the  square  root  of  IfJ. 

Operation. — \/Hi 

-_vioo_,„ 

V121 

EuLE. — Tlie  square  root  of  a  fraction  may  be  found  by 
extracting  the  square  root  of  the  numerator  and  denomina- 
tor separately. 

Mixed  numbers  may  be  reduced  to  the  decimal  form  before  ex- 
tracting the  root ;  or,  if  the  denominator  of  the  fraction  is  a  perfect 
square,  to  an  improper  fraction. 

In  extracting  the  square  root  of  a  number  containing  a  decimal^ 
begin  at  the  units'  place,  and  proceed  both  toward  the  left  and  the 
right  to  separate  into  periods,  then  proceed  as  in  the  extraction  of 
the  square  root  of  integers. 

Extract  the  square  root 


12. 

Of  iff. 

15. 

Of  .001225. 

18. 

Of  58.1406|» 

13. 

Ofi^T- 

16. 

Of  196.1369. 

19. 

Of  17f. 

14. 

OfyHir- 

17. 

Of  2.251521. 

20. 

Of  10795.21. 

L66 


EVOLUTIOK. 


21.  What  is  the  square  root  of  3486784401  ? 

22.  What  is  the  square  root  of  9.0000994009  ? 

23.  Find  the  value  of  32^  to  6  decimal  places. 

24.  Find  the  square  root  of  2f  to  4  decimal  places. 

25.  Find  the  square  root  of  f  to  5  decimal  places. 

26.  Find  the  value  of  .1254  to  5  decimal  places. 
Find  the  second  member  of  the  following  equations  : 

27.  a/3369  +  Vi296=  ?      |       28.  (36^)^  x  a/."25^=  ? 


29.  2.83 -^  a/.  11 7649  zz:? 

30.  vnu-^m^-^^=? 


31. 


9 


,i 


1=  X 


\/32        a/92 


32.  'v/^:6896  +  .3729  x|  of  a/.256=:  ? 

33.  (7.2  -  A/277or)'  -r-  (|f  =  ? 

34.  (a/8T—  16*)  X  (a/T69  +  25*)  =  ? 

35.  a/2642  X  4.41  -t-  (5.3361)*  -  (2.3^  x 


Geometrical  Explanation  of  Square  Eoot. 

814.  What  is  the  length  of  one  side  of  a  square  whose 
area  is  729  square  feet  ? 

Fia.  1.  Let  Fig.  1  represent  a  square  whose  area 

is  729  square  feet.     It  is  required  to  find  the 
length  of  one  side  of  this  square. 

Since  the  area  of  a  square  is  equal  to  the 
square  of  one  of  its  sides,  a  side  may  be 
found  by  extracting  the  square  root  of  the 
area. 

Since  729  consists  of  two  periods,  its  square 
root  will  consist  of  two  figures.     The  great- 
est number  of  tens  whose  square  is  con- 
tained in  700  is  2.     Hence  the  length  of  the  side  of  the  square  ia 
20  feet  plus  the  units'  figure  of  the  root. 


Fig.  2. 

^^^^^^1 

B 

jiiiii 

§H 

SQUARE     ROOT.  167 

Removing  the  square  whose  side  is  20  feet  and  whose  area  is  400 
square  feet,  there  remains  a  surface  whose 
area  is  329  square  feet  (Fig.  2).  ^i'his  re- 
mainder consists  of  two  equal  rectangles, 
each  of  which  is  20  feet  long,  and  a  square 
whose  side  is  equal  to  the  width  of  each 
rectangle..  The  units'  figure  of  the  root 
is  equal  to  the  width  of  one  of  these 
rectangles. 

The  area  of  a  rectangle  is  equal  to  the 
product  of  its  length  and  width  (462) ; 
hence,  if  the  area  be  divided  by  the  length, 

the  quotient  will  be  the  width.  Now,  since  the  two  rectangles 
contain  the  greater  portion  of  the  329  square  feet,  2  x  20  or  40, 
the  length  of  the  two  rectangles,  may  be  used  as  a  trial  divisor  to 
find  the  width.  Dividing  329  by  40,  the  quotient  is  8.  But  this 
quotient  is  too  large  for  the  width  of  the  rectangles,  for  if  8  feet  is 
the  width,  the  area  of  Fig.  2  will  be  40  x  8  +  8^  or  384  square  feet. 
Taking  7  feet  for  the  width  of  the  rectangles,  the  area  of  Fig.  2  is 
40  X  7  +  72  or  329  square  feet.  Hence  20  +  7  or  27  feet  is  the  length 
of  a  side  of  the  square  whose  area  is  729  square  feet. 

PROBIjEMS, 

815.  1.  A  square  field  contains  1016064  square  feet. 
What  is  the  length  of  each  side  ? 

2.  A  square  farm  contains  361  A.  Find  the  length  of 
one  side. 

3.  A  field  is  208  rd.  long  and  13  rd.  wide.  What  is  the 
length  of  the  side  of  a  square  containing  an  equal  area  ? 

4.  If  251  A.  65  P.  of  land  are  laid  out  in  the  form  of  a 
square,  what  will  be  the  length  of  each  of  its  sides  ? 

5.  A  circular  island  contains  21170.25  P.  of  land.  What 
is  the  length  of  the  side  of  a  square  field  of  equal  area  ? 

6.  If  it  cost  $312  to  enclose  a  field  216  rd.  long  and 
24  rd.  wide,  what  will  it  cost  to  enclose  a  square  field 
of  equal  area  with  the  same  kind  of  fence  ? 


168  EVOLUTION. 

CUBE    ROOT. 

816.  A  Perfect  Cube  is  a  number  which  has  an 
exact  cube  root. 

817.  The  Cube  Hoot  of  a  numbei*  is  one  of  the  three 
equal  factors  of  that  number.  Thus^  the  cube  root  of  125 
is  5,  since  5x5x5  =  125. 

818.  Pkii^ciples. — 1.  The  cuie  of  a  mimier  expressed 
by  a  single  figure  contains  no  figure  of  a  higher  order  than 
hundreds. 

2.  The  cube  of  tens  contains  no  significant  figure  of  a 
tower  order  than  thousands,  nor  of  a  higher  order  than 
hundred  thousands. 

3.  Tlie  cube  of  a  number  contains  three  time»  as  many 
figures  as  the  number,  or  three  times  as  many,  less  one  or 
two.    Thus, 

Pzzi              1  103=                          1^000 

3»  =           27  lOQS  =                1,000,000 

93  =         729  10003  =         1,000,000,000 

99s  =  907,299  100003  =  1,000,000,000,000 

4.  If  any  perfect  cube  be  separated  into  periods  of  three 
figures  each,  beginning  with  units'  place,  the  number  of 
periods  will  be  equal  to  the  number  of  figures  in  the  cube 
root  cf  that  number. 

WRITTEN     EXJEHC  I  S^S, 

819.  To  find  the  cube  root  of  a  number. 

1.  Find  the  cube  root  of  405224. 

OFERATIOKT. 

405,224  (  70  +  4  ^  74,  cube  root. 
70^  =  343  000 

702  X  3  =  14700 )  62  224 
743=  405  224 


CUBE     ROOT.  169 

Analysis. — Since  405224  consists  of  two  periods,  its  cube  root 
will  consist  of  two  figures  (818,  Prin.  4),  Since  224  cannot  be  a 
part  of  the  cube  of  the  tens  of  the  root  (818,  Prin.  2),  the  first 
figure  of  the  root  must  be  found  from  the  first  period,  405.  The 
greatest  number  of  tens  whose  cube  is  contained  in  405000  is  7. 
Subtracting  the  cube  of  7  tens  from  the  given  number,  the  remain- 
der is  62224.  This  remainder  is  equal  to  the  product  of  three  timea 
the  square  of  the  tens  of  the  root  by  the  units,  plus  three  times  the 
product  of  the  tens  by  the  square  of  the  units,  plus  the  cube  of  the 
units  (804,  Prin.)  But  the  product  of  the  square  of  tens  by  units 
cannot  be  of  a  lower  order  than  hundreds  (818,  Prin.  2) ;  hence 
the  number  represented  by  the  last  t\vo  figures,  24,  cannot  be  a  part 
of  three  times  the  product  of  the  square  of  the  tens  of  the  root  by 
the  units  ;  the  triple  product  must  therefore  be  found  in  the  part 
62200.  Hence,  if  62200  be  divided  by  3  x  70^  the  quotient,  which 
is  4,  will  be  the  units'  figure  of  the  root  or  a  figure  greater  than  the 
units'  figure.  Subtracting  74^  from  the  given  number,  the  result 
is  0  ;  hence  74  is  the  required  root. 

Instead  of  cubing  74,  the  parts  which  make  up  the  remainder 
62224  may  be  formed  and  added  thus  : 

3  X  702  X  4  =  58goO 
3  X  70  X  42  =    3360 

48  = 64 

62224/ 

Or,  since  4  is  a  common  factor  in  the  three  parts  which  make  up 
the  remainder,  these  parts  may  be  combined  thus  : 

3  X  702         =  14700 

3  X  70  X  4  =     840 

42= 16 

15556x4=62224. 

1  In  this  example,  14700  is  a  partial  or  trial  divisor,  and  15556  is 
a  complete  divisor, 

2.  If  the  cube  root  contains  more  than  two  figures,  it  may  be 
found  by  a  similar  process,  as  in  the  following  example,  where  it 
will  be  seen  that  the  partial  divisor  at  each  step  is  equal  to  thre© 
times  the  square  of  that  part  of  the  root  already  found. 


170 


EVOLUTION. 


2.  Find  the  cube  root  of  12812904. 


OPERATION. 


2003== 

1st  par.  divisor  3  x  200'      ==  120000 
3x200x30^  18000 

30^  =r      900 

1st  complete  divisor     138900 

3d  par.  divisor  3  x  280^  =  1 58700 
3  X  230  X  4.=    2760 

42=    16 


Cube  Root 
12,812,904  ( 200  +  30  +  4==234 
8  000  000 


4  812  904 
4167000 


645904 


645904 


2d  comilete  divisor  161476 

The  operation  may  be  abridged  as  follows  : 

12,812,904(234 
2^=      8 


1st  partial  divisor  3  X  20^     =1200 

3x20x3=  180 

3'^=      9 


1st  complete  divisor 


1389 


2d  par.  divisor 


3x230*     =158700 

3x230x4=    2760 

42=        16 


2d  complete  divisor 


161476 


4812 
4167 


645904 


645904 


EuLE. — I.  Separate  the  given  number  into  periods  of 
three  figures  each,  beginning  at  the  units'^  place. 

II.  Find  the  greatest  number  whose  cube  is  contained  in 
the  period  on  the  left ;  this  will  be  the  first  figure  in  the 
root.  Subtract  the  cube  of  this  figure  from  the  period  on 
the  left,  and  to  the  remainder  annex  the  next  period  to 
form  a  dividend, 

III.  Divide  this  dividend  by  the  partial  divisor,  ivhich 
is  3  times  the  square  of  the  root  already  found,  considered 
as  tens  ;  the  quotient  is  the  second  figure  of  the  root. 


CUBE     ROOT.  171 

IV.  To  the  partial  divisor  add  3  times  the  product  of  the 
second  figure  of  the  root  by  the  first  considered  as  tens,  also 
the  square  of  the  second  figure,  the  result  will  be  the  com- 
plete divisor. 

V.  Multiply  the  complete  divisor  by  the  second  figure  of 
the  root  and  subtract  the  product  from  the  dividend, 

VI.  //  there  are  more  periods  to  be  brought  down,  pro* 
ceed  as  before,  using  the  part  of  the  root  already  found, 
the  same  as  the  first  figure  in  the  previous  process. 

1.  If  a  cipher  occur  in  the  root,  annex  two  ciphers  to  the  trial 
divisor,  and  another  period  to  the  dividend  ;  then  proceed  as  before, 
annexing  both  cipher  and  trial  figure  to  the  root. 

2.  If  there  is  a  remainder  after  the  root  of  the  last  period  is  found, 
annex  periods  of  ciphers  and  proceed  as  before.  The  figures  of  the 
root  thus  obtained  will  be  decimals. 

What  is  the  cube  root 


3.  Of  15625  ? 

4.  Of  166375  ? 


5.  Of  1030301  ? 

6.  Of  4492125  ? 


7.  Of  1045678375  ? 

8.  Of  4080659192 ? 


9.  Find  the  cube  root  of  ^. 


Operation.— ^A  =  J^-A  =  |. 

EuLE. — The  cube  root  of  a  fraction  may  be  found  by 
extracting  the  cube  root  of  the  numerator  and  denominator. 

In  extracting  the  cube  root  of  decimal  numbers,  begin  at  the 
units'  place  and  proceed  both  toward  the  left  and  the  right,  to 
separate  into  periods  of  three  figures  each. 

Extract  the  cube  root 


14.  Of  .091125. 

15.  Of  12.812904. 


10.  Of  if^f         12.  Of  ^. 

11.  Of  flfff.       13.  Of  39304. 

16.  What  is  the  cube  root  of  98867482624  ? 

17.  What  is  the  cube  root  of  .000529475129  ? 

18.  Find  the  cube  root  of  ^  correct  to  4  decimal  places, 


172 


EVOLUTION. 


Find  the  second  member  of  the  following  equations 
19.     1.443+2.53==?  21. 

22. 


^0.  ^iifix^ifi==? 


V.4096  —  .2368  =  ? 
^^'54:872  — (21.952)*=? 


23.     24.8  +  v^l03.823  x  (.125)i  =  ? 
U.     V^166  -r-  \/6i  -  (4  X  ^^."512)  =  ? 

Geometrical  Explanation  of  Cube  Root. 

830.  What  is  the  length  of  the  edge  of  a  cube  whose 
volume  is  15625  cubic  feet  ? 

^i^-  1-  Let  Fig.  1  represent  a 

cube  whose  volume  is 
15625  cubic  feet.  It  is 
required  to  find  the  length 
of  the  edge  of  this  cube. 

Since  the  volume  of  a 
cube  is  equal  to  the  cube 
of  one  of  its  edges,  an 
edge  may  be  found  by 
extracting  the  cube  root 
of  the  volume. 

Since  15625  consists  of 
two  periods,  its  cube  root 
will  consist  of  two  figures.  The  greatest  number  of  tens  whose 
cube  is  contained  in  15000  is  2.  Hence,  the  length  of  the  edge  of 
the  cube  is  20  feet  plus  the  units'  figure  of  the  root.  Removing  the 
cube  whose  edge  is  20  feet  and  whose  volume  is  8000  cubic  feet, 
there  remains  a  solid  whose  volume  is  7625  cubic  feet  (Fig.  2). 
This  remainder  consists  of  solids  similar  to  those  marked  B,  C,  and 
D,  in  Fig.  1  and  Fig.  2  of  Art.  804. 

15,625(25 
2^=     8 

8  X  202        =  1200 
3  X  20  X  5  r=    300 

52=      25 


7625 


1525    7625 


CUBE     ROOT. 


173 


The  volume  of  a  rectan-  FiG.  2. 

gular  solid  is  equal  to  the 
product  of  the  area  of  its 
base  by  its  height  or  thick- 
ness (472) ;  hence,  if  the 
volume  be  divided  by  the 
area  of  the  base  the  quo- 
tient will  be  the  thickness. 
Now,  since  the  three  equal 
rectangular  solids,  each 
of  which  is  20  feet  square 
and  whose  thickness  is  the 
units'  figure  of  the  root, 
contain  the  greater  por- 
tion of  the  7625  cubic  feet,  3  x  20^  or  300  x  2^  may  be  used  as  a  trial 
divisor  to  find  the  thickness.  Dividing  7625  by  1200  the  quotient 
is  6.  But  this  quotient  is  too  large,  for  if  6  feet  is  the  thickness, 
the  volume  of  Fig.  2  will  be  3x202x6  +  3x20x62  +  63,  or  9576 
cubic  feet.  Taking  5  feet  for  the  thickness,  the  volume  of  Fig.  2 
is  7625  cubic  feet,  for  3  x202  x  5  +  3  x  20  x52  +  5'=:(300x  2^  +  30x2 
X  5  +  52)  5  =  1525  X  5=7625.  Hence,  25  feet  is  the  length  of  the  edge 
of  a  cube  whose  volume  is  15625  cubic  feet. 

PJtOBTjEMS, 

831.  1.  What  is  the  length  of  the  edge  of  a  cubical 
box  that  contains  46656  cu.  inches  ? 

2.  What  must  be  the  length  of  the  edge  of  a  cubical 
bin  that  shall  contain  the  same  volume  as  one  that  is 
16  ft.  long,  8  ft.  wide,  and  4  ft.  deep  ? 

3.  What  are  the  dimensions  of  a  cube  that  has  the 
same  volume  as  a  box  2  ft.  8  in.  long,  2  ft.  3  in.  wide,  and 
1  ft.  4  in.  deep  ? 

4.  How  many  square  feet  in  the  surface  of  a  cube 
whose  volume  is  91125  cubic  feet  ? 

5.  What  is  the  length  of  the  inner  edge  of  a  cubical 
bin  that  contains  150  bushels  ? 


174  EVOLUTION. 

6.  What  is  the  depth  of  a  cubical  cistern  that  holds 
200  barrels  of  water  ? 

7.  Find  the  length  of  a  cubical  vessel  that  will  hold 
4000  gallons  of  water. 

ROOTS  OF  HIGHER  DEGREE. 

833.  Any  root  whose  index  contains  no  other  factors 

than  2,  or  3,  may  be  extracted  by  means  of  the  square  and 

cube  roots. 

If  any  power  of  a  given  number  is  raised  to  any  required  power, 
the  result  is  that  power  of  the  given  number  denoted  by  the  pro- 
duct of  the  two  exponents.  (801.)  Conversely,  if  two  or  more 
roots  of  a  given  number  are  extracted,  successively,  the  result  is 
that  root  of  the  given  number  denoted  by  the  product  of  the  indices. 

1.  What  is  the  6th  root  of  2176782336  ? 

OPERATION.  Analysis. — The  index  of  the  re- 

V^2176782336  =  46656  ^^^^^^  ^^^*  is  6  =  2  x  3  ;  hence  ex- 

3 tract  the  square  root  of  the  given 

V  46656  =  36  number,  and  the  cube  root  of  this 

Or,  result,  which  gives  36  as  the  6th  or 

-^2176783336  =  1396  '^^li^ed  "^t-^    O^,  first  find  the 
cube  root  of  the  given  number,  and 

V  1296  =  36  then  the  square  root  of  the  result. 

KuLE. — Separate  the  index  of  the  required  root  into  its 
prime  factors,  and  extract  successively  the  roots  indicated 
hy  the  several  factors  obtained;  the  final  result  will  ie  the 
required  root. 

2.  What  is  the  4th  root  of  5636405776  ? 

3.  What  is  the  8th  root  of  1099511627776  ? 

4.  What  is  the  6th  root  of  25632972850442049? 

5.  What  is  the  9th  root  of  1.577635  ? 

For  further  practical  applications  of  Involution  and  Evolution, 
Bee  **  Mensuration." 


833.  An  Arithmetical  Progression  is  a  suc- 
cession of  numbers,  each  of  which  is  greater  or  less  than 
the  preceding  one  by  a  constant  difference. 

Thus,  5,  7,  9,  11,  13,  15,  is  an  arithmetical  progression. 

834.  The  Terms  of  an  arithmetical  progression  are 
the  numbers  of  which  it  consists.  The  first  and  last  terms 
are  called  the  Extremes^  and  the  other  terms  the  Means. 

835.  The  Common  Difference  is  the  difference 
between  any  two  consecutive  terms  of  the  progression. 

836.  An  Increasing  Arithmetical  Frogres- 
sion  is  one  in  which  each  term  is  greater  than  the  pre- 
ceding one. 

Thus,  1,  3,  5,  7,  9,  11,  is  an  increasing  progression. 

837.  A  Decreasing  Arithfuetical  Progres- 
sion is  one  in  which  each  term  is  less  than  the  preced- 
ing one. 

Thus,  15,  13,  11,  9,  7,  5,  3,  1,  is  a  decreasing  progression. 

838.  The  following  are  the  quantities  considered  in 
arithmetical  progression  and  the  abbreviations  used  for 
them: 


1.  The  first  term,  (a). 

2.  The  last  term,  (I). 


3.  The  common  diflFerence,  (d). 

4.  The  number  of  terms,      (n). 


5.  The  sum  of  all  the  terms,  («). 


176  PROGRESSIONS. 

WMITTE  N      EXEJRCISES, 

839.  To  find  one  of  the  extremes,  when  the  other 
extreme,  the  common  difference,  and  the  number 
of  terms  are  given. 

1.  The  first  term  of  an  increasing  progression  is  8^  the 
common  difference  5,  and  the  number  of  terms  20 ;  what 
is  the  last  term  ?  •  u» 

OPERATION.  Analysis. — The  2d  term  is  8  +  5; 

20 1  =  19  the  3d  term  is  8  +  (5  x  2)  the  4th  term 

r-Q r^       ^ ^  ^o J        is  8  +  (5  X  3) ;  and  so  on.     Hence  8  + 

ly  X  D  +  b  —  iOd  _  ^.       ^^g  ^  g^  ^j.  ^^3  .g  ^^^  2Q^^  ^^  ^^^^^  ^^^^ 

2.  The  last  term  of  an  increasing  progression  is  103, 
the  common  difference  5,  and  the  number  of  terms  20 ; 
what  is  the  first  term  ? 

OPERATION.  Analysis.— The  1st  term  must  be 

-  a  number  to  which,  if  19  x  5  be  added, 

"^  the  sum  shall  be  103  ;  hence,  if  19  x  5 

103  —  19x5=:8=:6]^     is  subtracted  from  103,  the  remainder 

is  the  first  term. 

3.  The  first  term  of  a  decreasing  progression  is  203, 
the  common  difference  5,  and  the  number  of  terms  40 ; 
what  is  the  last  term  ? 

4.  The  last  term  of  a  decreasing  progression  is  1,  the 
common  difference  2,  and  the  number  of  terms  9  ;  what 
is  the  first  term  ? 

EuLE.— I.  If  the  given  extreme  is  the  less,  add  to  it  the 
product  of  the  common  difference  by  the  number  of  terms 
less  one. 

II.  If  the  given  extreme  is  the  greater,  subtract  from  it 

the  product  of  the  common  difference  by  the  number  of 

terms  less  one, 
^  il  z=za  +  (n  —  1)  X  d. 

Formula.— i         ,       ;        ^i      j 
*  a=.l  —  {n  —  1)  X  d. 


PROGRESSIONS.  177 

5.  The  first  term  of  an  increasing  progression  is  5,  the 
common  difference  4,  and  the  number  of  terms  8 ;  what 
is  the  last  term  ? 

6.  The  first  term  of  an  increasing  progression  is  2,  and 
the  common  difference  3  ;  what  is  the  50th  term  ? 

7.  The  first  term  of  a  decreasing  progression  is  100, 
and  the  common  difference  7  ;  what  is  the  13th  term  ? 

8.  The  first  term  of  an  increasing  progression  is  f ,  the 
common  difference  |,  and  the  number  of  terms  20  ;  what 
is  the  last  term  ? 

830.  To  find  the  common  difference,  when  the 
extremes  and  number  of  terms  are  given. 

1.  The  extremes  of  a  progression  are  8  and  103,  and 
the  number  of  terms  20  ;  what  is  the  common  difference  ? 

OPERATION.  Analysis.— The  difference  between 

I QQ Q  _^  i  Q  __  K  __  ^       the  extremes  is  equal  to  the  product 

of  the  common  difference  by  the 
number  of  terms  less  one  (829) ;  hence  the  common  difference  is 
ff,or5. 

2.  The  extremes  of  a  progression  are  1  and  17,  and  the 
number  of  terms  9  ;  what  is  the  common  difference  ? 

EuLE. — Divide  the  difference  between  the  extremes  by 
the  number  of  terms  less  one. 

Formula. — d  =  -^^? . 
n  —  1 

3.  The  extremes  are  3  and  15,  and  the  number  of  terras 
7 ;  what  is  the  common  difference  ? 

4.  The  extremes  are  1  and  51,  and  the  number  of  terms 
76  ;  what  is  the  common  difference  ? 


178  PROGRESSIONS. 

5.  The  youngest  of  ten  children  is  8^  and  the  eldest  44 
years  old ;  their  ages  are  in  arithmetical  progression. 
What  is  the  common  difference  of  their  ages  ? 

6.  The  amount  of  $800  for  60  years,  at  simple  interest, 
is  $4160.     What  is  the  rate  per  cent.  ? 

7.  The  extremes  are  0  and  2^,  and  the  number  of  terms 
18  ;  what  is  the  common  difference  ? 

831.  To  find  the  humber  of  terms,  when  the  ex- 
tremes and  common  difference  are  given. 

1.  The  extremes  of  a  progression  are  8  and  103,  and 

the  common  difference  5  ;  what  is  the  number  of  terms  ? 

OPERATION.  Analysis. — The  difference  between  the 

-1  no        Q  _i_  K  ___  1  Q       extremes  is  equal  to  the  product  of  the 

1    *    OA  common    difference    by    the   number   of 

■^^  +  ^^^^~^       terms  less  one  (830) ;  hence  the  number 

of  terms  less  one  is  equal  to  -^/  or  19 ; 

therefore  19  + 1  or  20  is  the  number  of  terms. 

2.  The  extremes  of  a  progression  are  1  and  17,  and  the 
common  difference  2  ;  what  is  the  number  of  terms  ? 

EuLE. — Divide  the  difference  between  the  ecctremes  by 
the  common  difference,  and  add  one  to  the  quotient. 

FOEMULA. — 71  =  ~ — h  1. 

3.  The  extremes  are  5  and  75,  and  the  common  differ- 
ence is  5  ;  what  is  the  number  of  terms  ? 

4.  The  extremes  are  \  and  20,  and  the  common  differ- 
ence is  6|- ;  what  is  the  number  of  terms  ? 

5.  A  laborer  received  50  cents  the  first  day,  54  cents 
the  second,  58  cents  the  third,  and  so  on,  until  his  wages 
were  $1.54  a  day  ;  how  many  days  did  he  work  ? 

6.  In  what  time  will  $500,  at  7  per  cent,  simple  inter- 
est, amount  to  $885  ? 


PEOGRESSIONS.  179 

832.  To  find  the  sum  of  all  the  terms,  when  the 
extremes  and  the  number  of  terms  are  given. 

1.  The  extremes  of  an  arithmetical  progression  are  2 
and  14,  and  the  number  of  terms  is  5  ;  what  is  the  sum 
of  all  the  terms  ? 

Analysis. — The  common  dif- 
OPEKATION.  erence  is  found  to  be  3  (830) ; 

2_L.    5_L    «_l11j_14       hence    the    required    sum    is 

""i^TuT    o T    K        ..        ^q^^l  *o  2  +  5  +  8  +  11  +  14,  or 
,^14  +  11+   8+   5+   2       i4^^i^s^5^3^    Addingthe 

2  S  =16  +  16  +  16  +  16  +  16  corresponding  terms  of  these 

2  5  =  16  X  5  z=  (2  +  14)  X  5  ^^o  progressions,  we  have  2 

o    I    1 4^  times  the  sum  —  16  x  5  =  (2  + 

S  =■ X  5  =  40.  14)  X  5  ;    hence    the    sum    is 


2 


^  +  14      K         Al^ 

X  5  =  40. 


2 

2.  The  extremes  of  an  arithmetical  progression  are  5 
and  75,  and  the  number  of  terms  is  15  ;  what  is  the  sum 
of  all  the  terms  ? 

KuLE. — Multiply  the  sum  of  the  extremes  by  half  the 
number  of  terms. 

Formula. — s  =  -  x  {a  +  I). 

3.  The  extremes  are  4  and  40,  and  the  number  of  terms 
is  7  ;  what  is  the  sum  of  all  the  terms  ? 

4.  The  extremes  are  0  and  250,  and  the  number  of 
terms  is  1000  ;  what  is  the  sum  of  all  the  terms  ? 

5.  How  many  strokes,  beginning  at  1  o'clock,  does  the 
hammer  of  a  common  clock  strike  in  12  hours  ? 

6.  A  body  will  fall  16^  ft.  in  the  first  second  of  its 
fall,  48|^  ft,  in  the  second  second,  80^^^  ft.  in  the  third 
second,  and  so  on  ;  how  far  will  it  fall  in  one  minute  ? 


180  PROGRESSIONS. 

833.  A  Geometrleal  Progression  is  a  succes- 
sion of  numbers,  each  of  which  is  greater  or  less  than  the 
preceding  one  in  a  constant  rafio. 

Thus,  1,  3,  9,27,  81,  etc.,  is  a  geometrical  progression. 

834.  The  Terms  of  a  geometrical  progression  are 
the  numbers  of  which  the  progression  consists.  The  first 
and  last  terms  are  called  the  Extremes^  and  the  other 
terms  the  Means. 

835.  The  Ratio  of  a  geometrical  progression  is  the 
quotient  obtained  by  dividing  any  term  by  the  preceding 
one. 

836.  An  Increasing  Geometrical  Frogres' 
sion  is  one  in  which  the  ratio  is  greater  than  1. 

Thus,  1,  3,  4,  8,  16,  etc.,  is  an  increasing  progression. 

837.  A  Decreasing  Geometrical  JProgres-^ 

sioni&  one  in  which  the  ratio  is  less  than  1. 

Thus,  1,  J,  J,  J,  Y^^,  etc.,  is  a  decreasing  progression. 

838.  An  Infinite  Decreasing  Geometrical 
Progression  is  one  in  which  the  ratio  is  less  than  1, 
and  the  number  of  terms  infi7iite. 

Thus,  1,  i,  J,  J,  ^^,  7^,  ^,  and  so  on  is  an  infinite  decreasing 
progression. 

839.  The  following  are  the  quantities  considered  in 
geometrical  progression  : 


1.  The  first  term  {a), 

2.  The  last  term   (Z). 


3.  The  ratio  (r). 

4.  The  number  of  terms    (w). 


5.  The  sum  of  all  the  terms  {s). 


PROGRESSIONS.  181 

WRITTJEN     EXEJRC  IS  1SS. 

840.  To  find  one  of  the  extremes,  when  the  other 
extreme,  the  ratio,  and  the  number  of  terms  are 
given. 

1.  The  first  term  of  a  progression  is  2,  the  ratio  3,  and 
the  number  of  terms  10  ;  what  is  the  last  term  ? 

Analysis.— The  2d  term  is  3  x  3  ;  the  third 
3'*^  =19683  term  is  2x3x3  or  2x3'^;  the  4tli  term  is 

2  2x3^;   and  so  on.     Hence  the  10th  or  last 

oaoaa 7         ^^^'^  is  2  x  3®  or  39366. 

2.  The  last  term  of  a  progression  is  39366,  the  ratio  3, 
and  the  number  of  terms  10  ;  what  is  the  first  term  ? 

OPERATION.  Analysis.  —  The  first  term  must  be  a  num- 

39355  ber,  by  which  if  3^  be  multiplied  the  product 

—39—  =  2  =  a     •  siiall  be  39366  ;  hence,  if  39366  be  divided  by 
3®,  the  quotient  will  be  the  first  term. 

3.  The  first  term  of  a  progression  is  1,  the  ratio  |,  and 
the  number  of  terms  9  ;  what  is  the  last  term  ? 

EuLE. — I.  If  the  given  extreme  is  the  first  term,  rmdti- 
ply  it  hy  that  power  of  the  ratio  whose  exponent  is  one  less 
than  the  number  of  terms. 

II.  If  the  given  extreme  is  the  last  term,  divide  it  hy 
that  power  of  the  ratio  whose  exponent  is  one  less  than  the 
number  of  terms. 

FoRMULiE. — I  =:  ar"^'^ ;       a  =  — — : . 

4.  The  first  term  of  a  geometrical  progression  is  6,  the 
ratio  4,  the  number  of  terms  6  ;  what  is  the  last  term  ? 

5.  The  last  term  is  192,  the  ratio  2,  and  the  number  of 
terms  7 ;  what  is  the  first  term  ? 


182  PROGRESSIOl^S. 

6.  A  drover  bought  20  cows,  agreeing  to  pay  $1  for  the 
first,  $2  for  the  second,  $4  for  the  third,  and  so  on  ;  how 
much  did  he  pay  for  the  last  cow  ? 

7.  Find  the  amount  of  $250  for  4  years  at  6  per  cent, 
compound  interest. 

The  first  term  is  350,  the  ratio  1,06,  and  the  number  of  terms  5. 

8.  If  1  cent  had  been  put  at  interest  in  1634,  what 
would  it  have  amounted  to  in  the  year  1874,  if  it  had 
doubled  its  value  every  12  years  ? 

841.  To  find  the  ratio,  when  the  extremes  and 
the  number  of  terms  are  given, 

1.  The  first  term  is  2,  the  last  term  512,  and  the  num- 
ber of  terms  5  ;  what  is  the  ratio  ? 

OPEKATiON.  ANAiiYBTS. — ^If  the  4th  power  of  the 

.gig  =256  ratio  be  multiplied  by  2, the  product  will 

4/^—  _  .  _  be  512  (840);  henc6,  if  512  be  divided 

Vx55b  —  4  —  r        ^^  ^  ^^^  quotient,  256,  will  be  the  4th 

power  of  the  ratio.    Hence  the  ratio  is  the  4th  root  of  256,  or  4. 

2.  The  first  term  is  1,  the  last  term  ^-J-g-,  and  the  num- 
ber of  terms  9  ;  what  is  the  ratio  ? 

EuLE. — Divide  the  last  term  by  the  first,  and  extract 
that  root  of  the  quotient  whose  index  is  one  less  than  the 
number  of  terms. 

Formula* — r=  V -- 

^   a 

3.  The  first  term  is  8,  the  last  term  5000,  and  the  num- 
ber of  terms  5  ;  what  is  the  ratio  ? 

4.  The  first  term  is  .0112,  the  last  term  7,  and  the 
number  of  terms  5  ;  what  is  the  ratio  ? 

5.  The  first  term  is  ^,  the  last  term  15^,  and  the 
number  of  terms  7  ;  what  is  the  ratio  ? 


PROGBESSIOKS.  183 

84:3«  To  find  the  number  of  terms,  when  the 
extremes  and  the  ratio  are  ^ven, 

1.  The  extremes  are  2  and  512>  and  the  ratio  is  4 ;  what 
is  the  number  of  terms  ? 

OPERATION.         Analysis. — If  512  be  divided  by  2,  the  quotient, 

2  )  512        256,  will  be  that  power  of  the  ratio  whose  exponent 

~Tr^        is  one  less  than  the  number  of  terms  (841).     But 

256  is  the  4th  power  of  the  ratio  4 :  hence  the  nuTn- 

4   =  2o6       her  of  terms  S^^. 

2.  The  extremes  are  1  and  ^^,  and  the  ratio  is  ^ ;  what 
is  the  number  of  terms  ? 

Rule. — Divide  the  last  term  iy  the  first ;  then  the  expo- 
nent of  the  power  to  which  the  ratio  must  ie  raised  to  pro- 
duce the  quotient  is  one  less  than  the  number  of  terms. 

.       I 
Formula. — r"^'^  =  - . 
a 

3.  The  extremes  are  2  and  1458,  and  the  ratio  is  3  ; 
what  is  the  number  of  terms  ? 

4.  The  extremes  are  -^  and  -^,  and  the  ratio  2 ;  what 
is  the  number  of  terms  ? 

843.  To  find  the  sum  of  all  the  terms,  when  the 
extremes  and  the  ratio  are  given. 

1,  The  extremes  are  2  and  128,  and  the  ratio  is  4 ;  what 
is  the  sum  of  all  the  terms  ? 

OPERATION. 

(128x4)^2       510 

4-1         ~  -3-  -  17U  ^  5 

4:  8  =       8  +  32  + 128  +  512  Analysis.— Subtract  the  sum  from  4 

8  =  2  +  8  +  32  + 128 times  the  sum,  and  510  remains,  which 

3  «  =  512  —  2  =  510  is  3  times  the  sum  ;  hence,  ^^yOT  170, 

510  _  170  _  g.  is  the  sum. 


184  PROGBESSIONS. 

2.  The  extremes  are  1  and  -^,  and  the  ratio  is  | :  what 
IS  the  sum  of  all  the  terms  ? 

is=         i  +  i  +  i  +  iV  +  A 

B,VL^,— Multiply  the  last  term  by  the  ratio^  and  divide 
the  difference  between  the  product  and  the  first  term  by  the 
difference  between  1  and  the  ratio. 

Formula. — s  = ;-. 

r  —  1 

3.  The  extremes  are  3  and  384,  and  the  ratio  is  2  ;  what 
is  the  sum  of  all  the  terms  ? 

4.  The  extremes  are  4f  and  ;^,  and  the  ratio  is  | ; 
what  is  the  sum  of  all  the  terms  ? 

5.  What  is  the  sum  of  all  the  terms  of  the  infinite  pro- 
gression 8,  4,  2,  1,  1^,  i,  ....  ? 

The  last  term  of  this  progression  may  be  conceived  as  0. 

6.  What  is  the  sum  of  all  the  terms  of  the  infinite  pro- 
gression 1,  I,  :^,  ^,  -gL.,  .  .  .  .  ? 

7.  What  is  the  sum  otl+^  +  ^  +  ^,  etc.,  to  infinity? 

8.  The  first  is  7,  the  ratio  3,  and  the  number  of  terms 
4  ;  what  is  the  sum  of  all  the  terms  ? 

First  find  tlie  last  term  by  Art.  840. 

9.  A  drover  bought  10  cows,  agreeing  to  pay  $1  for  the 
first,  $2  for  the  second,  S4  for  the  third,  and  so  on  ;  what 
did  he  pay  for  the  10  cows  ? 

10.  If  a  man  were  to  buy  3  2  horses,  paying  2  cents  for 
the  first  horse,  6  cents  for  the  second,  and  so  on,  what 
would  they  cost  him  ? 


844.  An  Annuity  is  a  sum  of  money  payable  an- 
nually. The  term  is  also  applied  to  a  sum  of  money 
payable  at  any  equal  intervals  of  time. 

845.  A  Certain  Annuity  is  one  wbich  continues 
for  a  definite  period  of  time. 

846.  A  Perpetual  Annuity  or  Fe7^j)etuity 
is  one  which  continues  forever. 

847.  A  Contingent  Annuity  is  one  which  begins 
or  ends,  or  both  begins  and  ends,  on  the  occurrence  of 
some  specified  future  event  or  events. 

848.  An  Annuity  Forborne  or  in  A7*rears 
is  one  the  payments  of  which  were  not  made  when  due. 

849.  The  Amount  or  Final  Value  of  an  an- 
nuity is  the  sum  of  all  the  payments  increased  by  the 
interest  of  each  payment^ from  the  time  it  becomes  due 
until  the  annuity  ceases. 

850.  The  Present  Worth  of  an  annuity  is  such  a 
sum  of  money  as  will,  in  the  given  time,  and  at  the  given 
rate  per  cent.,  amount  to  the  final  value. 

851i  An  annuity  is  said  to  be  deferred  when  it  does 
not  begin  until  after  a  certain  period  of  time  ;  it  is  said 
to  be  reversionary  when  it  does  not  begin  until  after  the 
occurrence  of  some  specified  future  event,  as  the  death 
of  a  certain  person  ;  and  it  is  said  to  be  in  possession 
when  it  has  begun,  or  begins  immediately. 


186  AKKUITIES. 

ANNUITIES    AT    SIMPLE    INTEREST. 
853.  All  problems  in  annuities  at  simple  interest  may 
be  solved  by  combining  the  rules  in  Arithmetical  Pro- 
gression with  those  in  Simple  Interest. 

WMITTJEN    EXJSRC  ISBS, 

853.  1.  What  is  the  amount  of  an  annuity  of  $300  for 
5  years,  at  6  per  cent,  simple  interest  ? 

OPERATION. 

300  +  372  Analysis.— At  the  end  of  the  5th 
H X  5  =  1680       year  the  following  sums  were  due : 

The  5th  year's  payment  =  $300, 

The  4tli  year's  payment  ==  $300  +  $18  = 
The  3d  year's  payment   =  $300  +  $36  = 
The  2d.year's  payment   =  $300  +  $54  =  $354, 
The  1st  year's  payment  =  $300  +  $72  =:  $373. 

These  sums  form  an  arithmetical  progression,  in  which  the  first 
term  is  the  annuity,  $300,  the  common  difference  is  the  interest  of 
the  annuity  for  1  year,  and  the  number  of  terms  is  the  number  of 
years.  The  sum  of  all  the  terms  of  this  progression  is  $1680  (882), 
which  is  the  amount  of  the  annuity. 

2.  A  father  deposits  annually^for  the  benefit  of  his  son, 
beginning  with  his  tenth  birthday,  such  a  sum  that  on 
his  21st  birthday  the  first  deposit,  at  simple  int.,  amounts 
to  $210,  and  the  sum  due  his  son  is  $1860.  Find  the 
annual  deposit,  and  at  what  rate  per  cent,  it  is  deposited. 

OPERATION. 

6  X  (1st  term  +  210)  =  1860.     (833.) 

Hence,  1st  term  =  310  —  210  =  100  —  a. 

(210  -  100)  ^  {1^  -1)  =:i^  =  10  ^  d.     (830.) 

Analysis. — Here  $210,  the  first  deposit,  is  the  last  term  ;  12,  the 
number  of  deposits,  is  the  number  of  terms  ; 


AIS^NUITIES.  187 

and,  $1860,  the  final  value  of  the  annuity,  is  the  sum  of  all  thsy 
terms.  Using  the  principle  of  832,  we  find  the  first  term  to  bv. 
$100,  which  is  the  annual  deposit.  By  880,  the  common  dif 
ference  is  found  to  be  $10  ;  hence  10  per  cent,  is  the  required  rate. 

3.  What  is  the  amount  of  an  annuity  of  $150  for  b\ 
years,  payable  quarterly,  at  1-|  per  cent,  per  quarter  ? 

4.  What  is  the  present  worth  of  an  annuity  of  $300 
for  5  years,  at  6  per  cent.  ? 

5.  What  is  the  present  worth  of  an  annuity  of  $500 
for  10  years,  at  10  per  cent.  ? 

6.  In  what  time  will  an  annual  pension  of  $500  amount 
to  $3450,  at  6  per  cent,  simple  interest  ? 

7.  Find  the  rate  per  cent,  at  which  an  annuity  of  $6000 
will  amount  to  $59760  in  8  years,  at  simple  interest. 

8.  A  man  works  for  a  farmer  1  yr.  6  mo.,  at  $20  per 
month,  payable  monthly  ;  and  these  wages  remain  unpaid 
until  the  expiration  of  the  whole  term  of  service.  What 
is  due  the  workman,  allowing  simple  interest  at  6  per 
cent,  per  annum  ? 

ANNUITIES    AT    COMPOUND    INTEREST. 

854.  All  problems  in  annuities  at  compound  interest 
may  be  solved  by  combining  the  rules  in  Geometrical 
Progression  with  those  in  Compound  Interest. 

WBITTEN     EXBItCIS  ES, 

1.  What  is  the  amount  of  an  annuity  of  $300  for  5 
years,  at  6  per  cent,  compound  interest  ^ 

OPERATION.  Analysis.— At  the  end  of  the 

300x1.065—300        -,^0110       5th  year  the    following  sums 

' ttt: "^^^  ioyi.it)  J 

.06  8^6  <1^G  : 


188  AKKUITIES. 

The  5tli  year's  payment  =  |300, 

The  4th  year's  payment  +  interest  for  1  year  =  $300  x  1.06, 

The  3d  year's  payment  +  compound  int.  for  2  years  ~  $300  x  1.06^, 
The  2d  year's  payment  +  compound  int.  for  3  years  =  $oOO  x  1.06^ 
The  1st  year's  payment  +  compound  int.  for  4  years  =  $300  x  1.06*. 

These  sums  form  a  geometrical  progression,  in  which  the  first 
term  is  the  annuity,  $300,  the  ratio  is  the  amount  of  $1  for  1  year, 
and  the  number  of  terms  is  the  number  of  years.  The  sum  of  all 
the  terms  of  this  progression  is  $1691.13  (843),  which  is  the 
amount  of  the  annuity. 

2.  What  is  the  present  worth  of  an  annuity  of  $300  for 
5  years,  at  6  per  cent,  compound  interest  ? 

OPERATION..  Analysis. — The  amount  of  this  an- 

1691.13  nuity  is  $1691.13.     The  amount  of  $1  for 

=  1363.71       5  years,  at  6  percent,  compound  interest, 
is  $1.338226  (587).    Hence  the  present 


1.338226 


worth  of  the  annuity  is  -f^^'  or  $1263.71. 

3.  Find  the  annuity  whose  amount  for  25  years,  at  6 
per  cent,  compound  interest,  is  $16459.35. 

4.  What  is  the  present  worth  of  an  annuity  of  $700 
for  7  years,  at  6  per  cent,  compound  interest  ? 

5.  An  annuity  of  $200  for  12  years  is  in  reversion  6 
years.  What  is  its  present  worth,  compound  interest 
at  6^? 

6.  A  man  bought  a  tract  of  land  for  $4800,  which  was 
to  be  paid  in  installments  of  $600  a  year ;  how  much 
money,  at  6  per  cent,  compound  interest,  would  discharge 
the  debt  at  the  time  of  the  purchase  ? 

7.  What  is  the  present  value  of  a  reversionary  lease  of 
$100,  commencing  14  years  hence,  and  to  continue  20 
years,  compound  interest  at  5  per  cent.  ? 


REVIEW.  189 

855.  SYNOPSIS  FOE  EEVIEW. 

[    Defs    i  ^'  ^Pow^^-  2-  Involution.  3.  Base,  or  Root.  4.  Ex- 
I      pouent.  5.  Square.  6.  Cube.  7.  Perfect  Power. 


2.  Principle. 

3.  802.     Rule.     1.  For  Integers.    2.  For  Fractions. 

4.  803,     1-  Principle.    2.  Geometrical  Illustration. 

5.  804.     1 .  Principle.    2.  Geometrical  Illustration. 

1    Defs  i  ^'  ^^^^^^  Root.    2.  Cube  Root,  etc.    3.  Evolution. 
(      4.  Radical  Sign.     5.  Index. 

2.  810.  Rule. 

3.  812.  Principles,  1,  2,  3,  4 

4.  813.  Rule,  I,  II,  III.    For  Fractions. 

5.  814:.  Geometrical  Illustration. 

6.  818.  Principles,  1,  2,  3,  -4. 

7.  819.  Rule,  I,  II,  III,  IV,  V,  VI.    For  Fractions. 

8.  820.  Geometiical  Illustration. 

9.  822.  Roots  of  a  Higher  Degree.    Bide. 

{ 1.  Arithmetical  Progression.    2.  Terms.    3.  Common 
^  1.  Defs.  -j     Difference,  4.  Increasing  Arithmetical  Progression. 
i     5.  Decreasing  Arithmetical  Progression, 

2.  Quantities  considered. 

3.  829.    Rule,  I,  II.    Formulm. 

4.  830.     Rule.    Formula. 

5.  831.     Rule.    Formvla. 

6.  832.    Rule.    Formula. 

( 1.  Geometii;Cal  Progression.     2.  Terms.      3.  Batio. 

1.  Defs.  ■<     4.  Increasing  Geom.  Prog.    5.  DecreoMng  Oeom, 

\     Prog.    6.  Infinite  Decreasing  Geom.  Prog. 

2.  Quantities  considered. 

3.  840.     Rule,  I,  II.    Formulm, 

4.  841.     Rule.    Formula. 

5.  842.    Rule.    Formula. 

6.  843.    Rule.    Formula. 
1.  Annuity.    2.  Certain  Annuity.     3.  Perpetuity. 

4.  Contingent  Annuity.     5.  Annuity  in  Arrears. 
6.  Amount.     7.  Present  Wortli  of  an  Annuity. 
8.  Deferred  Annuity.     9.  Reversionary  Annuity. 
10.  Annuity  in  Possession. 
3.  Annuities  at  Simple  Interest.  )  „    , ,         ,  ,     . 

^  2.  Annuities  at  Comf.  Interest.  |  ^^^1^°^'  how  solved. 


1.  Defs.  < 


85G.  Mensuration  is  the  process  of  finding  the  number  of 
units  in  extension. 

LINES. 

857.  A  Straight  Line  is  a  line  that 
does  not  change  its  direction.     It  is  the  short- 
est distance  between  two  points. 

858.  A*  Curved  lAne  changes  its  direc- 
tion at  every  point. 

859.  JFarallel  Lines  have  the  same 
direction  ;  and  being  in  the  same  plane  and 
equally  distant  from  each  other,  they  can  never 
meet. 

860.  A  HoHzonfal  Line  is  a  line  par' 
allel  either  to  the  horizon  or  water  level. 

801.  A  JPerpendirtilar  Line  is  a 
straight  line  drawn  to  meet  another  straight 
line,  so  as  to  incline  no  more  to  the  one  side 
than  to  the  other. 

A  perpendicular  to  a  horizontal  line  is  called  a  vertir 
ca^  lina 

ANGLES. 

862.  An  Angle  is  the  difference  in  the 
direction  of  two  lines  proceeding  from  a  com- 
mon point,  called  the  vertex. 

An^es  are  measured  by  degrees.    (301.) 

863.  A  Might  Angle  is  an  angle  formed 
by  two  lines  perpendicular  to  each  other. 

864:.  An  Obtuse  Angle  is  greater  than 
a  right  angle. 

865.  An  Acute  Angle  is  less  than  a 
right  angle, 
except  rigbt  angles  are  caQed  obHgtte  (mgke. 


Horizontal. 


TKIAKGLES. 


191 


PLANE  FIGURES. 

866.  A  Platte  Figure  is  a  portion  of  a  plane  surface  bounded 
by  straight  or  curved  lines. 

867.  A  Polygon  is  a  plane  figure  bounded  by  straight  lines. 

868.  The  Perhneter  of  a  polygon  is  the  sum  of  its  sides. 

869.  The  Area  of  a  plane  figure  is  the  surface  included 
within  the  lines  which  bound  it.    (460.) 

A  regular  polygon  has  all  its  sides  and  all  its  angles  equal. 

The  altitude  of  a  polygon  is  the  perpendicular  distance  between  its  hose  and  ^ 
side  or  angle  opposite. 

A  polygon  of  three  sides  is  called  a  trigon,  or  triangle ;  of  four  sides»  a  tetror 
gon^  or  quadrilateral;  of  five  sides,  &  pentagon^  etc. 


Pentagon.       Hexagon.        Heptagon.        Octagon.         Nonagon.         Decagon. 


TRIANGLES. 

870.  A  Triangle  is  a  plane  figure  bounded  by  three  sides, 
and  having  three  angles. 

871.  A  night- Angled  Triangle 

is  a  triangle  having  one  right  angle. 

872.  The  Hypothenuse  oi2iTLg\ii' 
angled  triangle  is  the  side  opposite  the 
right  angle. 

873.  The  Base  of  a  triangle,  or  of 

any  plane  figure,  is  the  side  on  which  it  may  be  supposed  to  stand. 

874.  The  Perpendicular  of  a  right-angled  triangle  is  the 
side  which  forms  a  right  angle  with  the  base. 

875.  The  Altitude  of  a  triangle  is  a  line  drawn  from  the  angle 
opposite  perpendicular  to  the  base. 

1.  The  dotted  lines  in  the  following  figures  represent  the  altitude. 

2.  Triangles  are  named  from  the  relation  both  of  their  sides  and  angles. 


192  MENSURATION. 

876.  An  Equilateral  Triangle  has  its  three  sides  equal. 

877.  An  Isosceles  Tri'angr^<?  has  only  two  of  its  sides  equal. 

878.  A  Scalene  Triangle  has  all  of  its  sides  unequal. 
Fig.  1.  FiQ.  2.  Fio.  3. 


Equilateral.  Isosceles.  Scalene. 

879.  An  Equiangular*  Triangle  has  three  equal  angles 
(Fig.  1.) 

880.  An  Acute-angled  Triangle  has  three  acute  anglea 
(Fig.  2.) 

881.  An  Obtuse-angled  Triangle  has  one  obtuse  angle. 
(Fig.  3.) 

SS^2.  The  base  and  altitude  of  a  triangle  being 
given  to  find  its  area, 

1.  Find  the  area  of  a  triangle  whose  base  is  26  ft.  and  altitude 
14.5  feet. 

145 

Operation.— 14.5  x 26^2=rl88Jsq.ft.   Or,  26  x -^=188^square 

feet,  area. 

2.  What  is  the  area  of  a  triangle  whose  altitude  is  10  yards  and 
base  40  feet  ? 

Rule.— 1.  Divide  the  product  of  the  base  and  altitude  hy  2.    Or, 

2.  Multiply  the  base  by  ona-half  the  altitude. 

Find  the  area  of  a  triangle 

3.  Whose  base  is  13  ft.  6  in.  and  altitude  6  ft.  9  in. 

4.  Whose  base  is  25.01  chains  and  altitude  18.14  chains. 

5.  What  is  the  cost  of  a  triangular  piece  of  land  whose  base  is 
15.48  ch.  and  altitude  9.67  ch.,  at  $60  an  acre? 

6.  At  $.40*  a  square  yard,  find  tho  cost  of  paving  a  triangular 
court,  its  base  being  105  feet,  and  its  altitude  21  yards  ? 

7.  Find  the  area  of  the  gable  end  of  a  house  that  is  28  ft.  wide, 
and  the  ridge  of  the  roof  15  ft.  higher  than  the  foot  of  the  rafters. 


TRIANGLES.  193 

883.  The  area  and  one  dimension  being  given  to 
find  the  other  dimension. 

1.  What  is  the  base  of  a  triangle  whose  area  is  189  square  feet 
and  altitude  14  feet  ? 

Operation.— (189  sq.  ft.  x  2)-t-14  =  27  ft.,  hose, 

2.  Find  the  altitude  of  a  triangle  whose  area  is  20J  square  feet 
and  base  3  yards. 

Rule. — Double  the  area,  then  divide  hy  the  given  dimension. 

Find  the  other  dimension  of  the  triangle 

3.  When  the  area  is  65  sq.  in.  and  the  altitude  10  inches. 

4.  When  the  base  is  42  rods  and  the  area  588  sq.  rods. 

5.  When  the  area  is  6  J  acres  and  the  altitude  17  yards. 

6.  When  the  base  is  12.25  chains  and  the  area  5  A.  33  P. 

7.  Paid  $1050  for  a  piece  of  land  in  the  form  of  a  triangle,  at  the 
rate  of  $5  J  per  square  rod.     If  the  base  is  8  rd.,  what  is  its  altitude  ? 

884.  The  three  sides  of  a  triangle  being  given  to 
find  its  area. 

1.  Find  the  area  of  a  triangle  whose  sides  are  30,  40,  and  50  ft. 

Operation.— (30  +  40  +  50)-j-2  =  60 ;  60-30  =  30  ;  60-40  =  20  ; 
60-50  =  10.     /v/60x  30x20x10  =  600  ft.,  area. 

2.  What  is  the  area  of  an  isosceles  triangle  whose  base  is  20  ft., 
and  each  of  its  equal  sides  15  feet  ? 

BjjuR.—From  half  the  svm  of  the  three  sides,  subtract  each  side 
separately  ;  multiply  the  half-%um  and  the  three  remainders  together; 
the  square  root  of  the  product  is  the  a/rea, 

3.  Find  the  area  of  a  triangle  whose  sides  are  25,  36,  and  49  in. 

4.  How  many  acres  in  a  field  in  the  form  of  an  equilateral  tri- 
angle whose  sides  each  measure  70  rods  ? 

ff.  The  roof  of  a  house  30  ft.  wide  has  the  rafters  on  one  side 
20  ft.  long,  and  on  the  other  18  ft.  long.  How  many  square  feet  of 
boards  will  be  required  to  board  up  both  gable  ends  ? 


194 


MEKSUEATIOK. 


885.  The  following  principles  relating  to  right-angled  triangles 

have  been  established  by  Geometry  ; 

Principles. — 1.  The  square  of  the 
hypothenuae  of  a  right-angled  triangle 
is  equal  to  the  sum  of  the  squares  of 
the  other  two  sides. 

2.  2^he  square  of  the  base,  or  of  the 
perpendicular,  of  a  right-angled  tri- 
angle is  equul  to  the  square  of  the 
hypothenuse  diminished  by  the  square 
of  the  other  side, 

886.  To  find  the  hypothenuse. 

1 .  The  base  of  a  right-angled  triangle  is  13,  and  the  perpendicu- 
lar 16.    What  is  the  length  of  the  hypothenuse  ? 

Operation.— 122  + 16^  =  400  (Prin.  1).    ^^  =  ^^^  hypotJienuae, 

2.  The  foot  of  a  ladder  is  15  feet  from  the  base  of  a  building,  and 
the  top  reaches  a  window  36  feet  above  the  base.  What  is  the 
length  of  the  ladder  ? 

Rule. — Extract  the  square  root  of  the  sum  of  the  squares  of  the 
base  and  the  perpendicular  ;  the  result  is  the  hypothenuse, 

3.  If  the  gable  end  of  a  house  40  ft.  wide  is  16  ft.  high,  what  is 
the  length  of  the  rafters  ? 

4.  A  park  25  chains  long  and  23  chains  wide  has  a  walk  running 
through  it  from  opposite  corners  in  a  straight  line.  What  is  the 
length  of  the  walk  ? 

5.  A  room  is  20  ft.  long,  16  ft.  wide,  and  12  ft.  high.  What  is  the 
distance  from  one  of  the  lower  corners  to  the  opposite  upper  corner  t 

887.  To  find  the  base  or  perpendicular, 

1.  The  hypothenuse  of  a  right-angled  triangle  is  35  feet,  and  the 
perpendicular  28  feet.     Find  the  base. 

Operation.— 352  -  28"  =  441  (Prin.  2).    y'Sl  =  21  ft.,  base. 


QUADRILATERALS. 


195 


2.  The  hypothenuse  of  a  right-angled  triangle  is  53  yards  and 
the  base  84  feet.     Find  the  perpendicular. 

Rule. — Extract  the  square  root  of  the  difference  between  the  square 
of  the  hypothenuse  and  the  square  of  the  given  side;  the  result  is  the 
required  side. 

3.  Find  the  width  of  a  house,  whose  rafters  are  13  ft.  and  15  ft. 
long,  and  that  form  a  right  angle  at  the  point  in  which  they  meet. 

4.  A  line  reaching  from  the  top  of  a  precipice  120  feet  high,  on 
the  bank  of  a  river,  to  the  opposite  side  is  380  feet  long.  How 
wide  is  the  river  ? 

5.  A  ladder  52  ft.  long  stands  against  the  side  of  a  building. 
How  many  feet  must  it  be  drawn  out  at  the  bottom  that  the  top 
may  be  lowered  4  feet  ? 

QUADRILATERALS. 

888.  A  Quadrilateral  is  a  plane  figure  bounded  by  four 
straight  lines. 

There  are  three  kinds  of  quadrilaterals,  the  ParaUdogram,  Trapezoid^  and 
Trapezium. 

889.  A  Parallelogram  is  a  quadrilateral  which  has  its 
opposite  sides  parallel. 

There  are  four  kinds  of  parallelograms,  the  Square,  Bectangle,  Bhombcid,  and 
Bhomlus, 

890.  A  Hectangle  is  any  parallelogram  having  its  angles 
right  angles. 

891.  A  Square  is  a  rectangle  whose  sides  are  equal. 

892.  A  Rhomboid  is  a  parallelogram  whose  opposite  sides 
only  are  equal,  and  whose  angles  are  not  right  angles. 

893.  A  Hhombtis  is  a  parallelogram  whose  sides  are  all 
equal,  but  whose  angles  are  not  right  angles. 


Square. 


Rectangle. 


Rhomboid. 


Rhombus. 


196 


M  E  i^  S  U  R  A  T  I  0  N  . 


894.  A  Trapezoid  is  a  quadrilateral,  two  of  whose  sides  are 
parallel. 

895.  A  Trapezium  is  a  quadrilateral  having  no  two  sides 
parallel. 

896.  The  Altitude  of  a  parallelogram  or  trapezoid  is  the  per 

pendicular  distance  between  its  parallel  sides. 

The  dotted  vertical  lines  in  the  figure  represent  the  altitude. 

897.  A  lyiagonal  of  a  plane  figure  is  a  straight  line  joining 
the  vertices  of  two  angles  not  adjacent. 


Parallelogram. 


Trapezoid. 


Trapezium. 


1*12  O  B  L  EMS  . 

898.  To  find  the  area  of  any  parallelogram. 

1.  Find  the  area  of  a  parallelogram  whose  base  is  16.25  feet  an'* 
altitude  7.5  feet. 

Operation.— 16.25  ft.  x  7.5  =  121.875  sq.  feet,  area. 

2.  The  base  of  a  rhombus  is  10  feet  6  inches,  and  its  altitude 
8  feet.     What  is  its  area  ? 

Rule. — Multiply  the  base  by  the  altitude. 

3.  How  many  acres  in  a  piece  of  land  in  the  form  of  a  rhomboid, 
the  base  being  8.75  ch.  and  altitude  6  chains  ? 

899.  To  find  the  area  of  a  trapezoid. 

1.  Find  the  area  of  a  trapezoid  whose  parallel  sides  are  23  and 
11  feet,  and  the  altitude  9  feet. 


Operation.— 23  ft.  + 11  ft. -^2=17  ft. ;  17  ft.  x  9=153  sq.  ft.,  area. 

2.  Required  the  area  of  a  trapezoid  whose  parallel  sides  are  178 
and  146  feet,  and  the  altitude  69  feet. 

Rule, — Multiply  one- half  the  sum  of  the  parallel  sides  by  the 
altitude. 


0IECLES.  19? 

3.  How  many  square  feet  in  a  board  16  ft.  long,  18  inches  wide 
at  one  end  and  25  inches  wide  at  the  other  end  ? 

4.  One  side  of  a  quadrilateral  field  measures  38  rods  ;  the  side 
opposite  and  parallel  to  it  measures  26  rods,  and  the  distance  be- 
tween the  two  sides  is  10  rods.     Find  the  area. 

900.  To  find  the  area  of  a  trapezium. 

1.  Find  the  area  of  a  trapezium  whose 
diagonal  is  42  feet  and  perpendiculars  to  this 
diagonal,  as  in  the  diagram,  are  16  feet  and 
18  feet. 


Operation.— (18  ft.  + 16  ft.  -^2)  x  42  =  714  sq.  feet,  area, 

2.  Find  the  area  of  a  trapezium  whose  diagonal  is  35  ft.  6  in.,  and 
the  perpendiculars  to  this  diagonal  9  feet  and  3  feet. 

Rule. — Multiply  the  diagonal  hy  half  the  sum  of  the  perpendicu- 
lars drawn  to  it  from  ihe  vertices  of  opposite  angles, 

3.  How  many  acres  in  a  quadrilateral  field  whose  diagonal  is 
80  rd.  and  the  perpendiculars  to  this  diagonal  20.453  and  50.832  rd.  ? 

To  find  the  area  of  any  regular  polygon,  multiply  its  perimeter,  or  the  Bnm  of 
ita  sides,  by  one-half  the  perpendicular  falling  from  its  center  to  one  of  its  sides. 

To  find  the  area  of  an  irregular  polygon,  divide  the  figure  into  triangles  and 
trapeziums,  and  find  the  area  of  each  separately.  The  sum  of  these  areas  will 
be  the  area  of  the  whole  polygon. 


THE  CIRCLE. 

901.  A  Circle  is  a  plane  figure  bounded  by  a  curved  line, 
called  the  circumference ^  every  point  of  which  is 
equally  distant  from  a  point  within  called  the 
center, 

902.  The  Diameter  of  a  circle  is  a  line 
passing  through  its  center,  and  terminated  at  both 
ends  by  the  circumference. 

903.  The  Ttaditis  of  a  circle  is  a  line  extending  from  its  cen- 
ter to  any  point  in  the  circumference.     It  is  one-half  the  diameter. 


198  MENS  U  RATIO  K. 

rS OB  LE  M  S  » 

904.  Wlien  either  the  diameter  or  the  circum- 
ference of  a  circle  is  given,  to  find  the  other  di- 
mension of  it. 

1.  Find  the  circumference  of  a  circle  wliose  diameter  is  20  inches. 
Operation.— 20  in.  x  3.1416  =  62.832  in.  =  5  ft.  2.832  in.,  ci/rcum. 

2.  Find  the  diameter  of  a  circle  whose  circumference  is  62.832  ft. 
Operation.— 62.832  ft.-f-3.1416  =  20  ft.,  diameter, 

3.  Find  the  diameter  of  a  wheel  whose  circumference  is  50  feet. 

Rule. — 1.  Multiply  the  diameter  by  3.1416  ;  the  product  is  the  oir- 
cumference. 
2.  Bimde  the  circumferenee  by  3.1416  ;  the  quotient  is  the  diameter. 

4.  What  is  the  diameter  of  a  tree  whose  girt  is  18  ft.  6  in.  ? 

5.  What  is  the  radius  of  a  circle  whose  circumference  is  31.416  ft.? 

6.  Find  the  circumference  of  the  greatest  circle  that  can  be 
drawn  with  a  string  14  inches  long,  used  as  a  radius. 

905.  To  find  the  area  of  a  circle,  wlien  both  its 
diameter  and  circumference  are  given,  or  when 
eitlier  is  given. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  10  feet  and  cir- 
cumference 31.416  feet?  t 


Operation.— 31.416  ft.  x  IOh-4  =  78.54  sq.  ft.,  area, 

2.  Find  the  area  of  a  circle  whose  diameter  is  10  feet. 
Operation.— 10  ft.^  x  .7854  =  78.54  sq.  feet,  a/rea, 

3.  Find  the  area  of  a  circle  whose  circumference  is  31.416  feet. 
Operation.— 31.416  ft.-^3.l416=rlO  ft.,  diam,;  (10  ft.j^x  .7854= 

78.54  sq.  feet,  a^ea. 

Rules. — To  find  the  area  of  a  circle  : 

1.  Multiply  i  of  its  diameter  by  the  circumference, 

2.  Multiply  the  square  of  its  diameter  by  .7854. 

4.  What  is  the  area  of  a  circular  pond  whose  circumference  is 
200  chains 

5.  The  distance  around  a  circular  park  is  1 J  miles.     How  many 
acres  does  it  contain  ? 


CIRCLES.  199 

906.  To  find  the  diameter  or  the  circumference 
of  a  circle,  when  the  area  is  g^iven. 

1.  What  is  tlie  diameter  of  a  circle  whose  area  is  1319.472  ? 


Operation.— 1319.472^.7854  =z  1689  ;  \/lQSO  =  40.987 + ,  diam- 
eter. 

2.  What  is  the  circumference  of  a  circle  whose  area  is  19.635  ? 

Opekation— 19,635  -i-  3.1416  =  6,25  ;  ^"6^5=12.5,  racUiis;  2.5  x 
2  X  3.1416  =  15.708,  circumference. 

Rule.— 1.  Divide  the  area  by  .7854  and  extract  the  square  root  of 
the  quotient ;  the  resuU  is  the  diameter, 

2.  Divide  the  area  hy  3.1416  and  extract  the  square  root  of  ths 
quotient ;  the  result  is  the  radius^  The  cireumference  is  obtained  by 
Art.  904,  1.  Or, 

8.  Divide  the  area  by  .07958  and  find  the  square  root  of  the  quotient. 

3.  The  area  of  a  circular  lot  is  38.4846  square  rods.  What  is  its 
diameter  ? 

4.  The  area  of  a  circle  is  286.488  equare  feet.  Required  the 
diameter  and  the  circumference. 

907.  To  find  the  side  of  an  inscribed  square  when 
the  diameter  of  the  circle  is  known. 

1.  What  is  the  side  of  a  square  inscribed  in  a 
circle  whose  diameter  is  6  rods  ? 

Operation. — 6^  -f-  2  =  18 ;  ,y^l8— 4.24  rods,  side 
ofsquAire. 

2.  The  diameter  of  a  circle  is  200  feet.    Find 
the  side  of  the  inscribed  square. 

Rule.— 1.  Extract  the  square  root  of  half  the  square  of  tlie  diam  | 
t^er.    Or, 

2.  Multiply  the  diameter  by  .7071. 

3.  The  circumference  of  a  circle  is  104  yards.  Elnd  the  side  of 
the  inscribed  square. 

4.  The  area  of  a  circle  is  78.54  square  feet  Find  the  side  of  the 
inscribed  square. 


200  MEKSURATIOK. 

908.  To  find  the  area  of  a  circular  ring   formed 
by  two  concentric  circles. 

1.  Find  the  area  of  a  circular  ring,  when 
the  diameters  of  the  circles  are  20  and  80  feet. 


Opekation.— (30  +  20  X  30  -  20)  x  .7854  = 
392.7  sq.  ft.,  area, 

2.  Find  the  area  of  a  circular  ring  formed 
by  two  concentric  circles,  whose  diameters  are 
7  ft.  9  in.  and  4  ft.  3  in. 
Rule. — MulMpiy  the  mm  of  the  two  diameters  by  their  difference, 
and  the  product  by  .7854 ;  the  result  is  the  area. 

3.  Two  diameters  are  35.75  and  16.25  ft. ;  find  the  area  of  the  ring. 

4.  The  area  of  a  circle  is  1  A.  154.16  P.     In  the  center  is  a  pond  of 
water  10  rd.  in  diameter  ;  find  the  area  of  the  land  and  of  the  water. 

909.  To  find  a  mean  proportional  between  two 
numbers. 

1.  What  is  a  mean  proportional  between  3  and  12  ? 


Operation. — '\/l2  x  3  =  6,  the  mean  proportional. 

When  three  nnmhers  are  proportional^  the  product  of  the  extremes  is  equal  to 
the  square  of  the  mean. 

EuLK — Extract  the  square  root  of  the  product  ofth^e  two  numbers. 

Find  a  mean  proportional  between 

2.  42  and  168.        |        3.  64  and  12.25.         |        4.  |f  and  /y. 

5.  A  tub  of  butter  weighed  36  lb/  by  the  grocer's  scales  ;  but 
weighing  it  in  the  other  scale  of  the  balance,  it  weighed  only  30 
pounds.    What  was  the  true  weight  of  the  butter  ? 

SIMILAR   PLANE   FIGURES. 

910.  Similar  I^lane  Fignres  are  such  as  have  the  same 
form  J  viz.,  equal  angles,  and  their  like  dimensions  proportional. 

All  circles,  squares,  equiangular  triangles,  and  regular  polygons  of  the  same 
number  of  sides  are  similar  fignres. 
The  like  dimensions  of  circles  are  their  radii,  diameters,  and  circumferences. 

Principles. — 1.  The  like  diwjensions  of  similar  plane  figures  are 
proportional. 


SIMILAR     PLANE     FIGURES.  201 

2.  The  areas  of  dmilar  plane  figures  are  to  each  other  as  the  squa/res 
of  their  like  dimensions.     And  conversely, 

3.  The  like  dimensions  of  similar  plane  figures  are  to  each  other  aa 
the  square  roots  of  their  areas. 

The  same  principles  apply  also  to  the  surfaces  of  all  similar  figures,  such  as 
triangles,  rectangles,  etc. ;  the  surfaces  of  similar  so/erfs,  as  cubes,  pyramids,  etc.; 
and  to  similar  curved  surfaces,  as  of  cylinders,  cones,  and  spheres.    Hence, 

4.  The  surfaces  of  all  similar  figures  are  to  each  other  as  the  squa/res 
of  their  like  dimensions.    And  conversely, 

5.  Their  dimensions  are  as  the  square  roots  of  their  surfaces. 

rR  OBLEM  S, 

1.  A  triangular  field  whose  base  is  12  ch.  contains  2  A.  80  P. 
Find  the  area  of  a  field  of  similar  form  whose  base  is  48  chains. 

Operation.— 122 :  48^ : :  2  A.  80  P. :  a?  P. =6400  P.  =  40  A.,  area. 
(Prin.  2.) 

2.  The  side  of  a  square  field  containing  18  acres  is  60  rods  long. 
Find  the  side  of  a  similar  field  that  contains  J  as  many  acres. 

Operation.— 18  A. :  6  A.  : :  60"^  :  ic^  =1200  ;  ^1200  =  34.64  rd.  +  , 
side,    (Prin.  3.) 

3.  Two  circles  are  to  each  other  as  9  to  16  ;  the  diameter  of  the 
less  being  112  feet,  what  is  the  diameter  of  the  greater? 

Operation— 9  :  16  : :  112^ :  a;^  =  3  :  4  : :  112  :  a;  =  149  ft.  4  in., 
diameter.    (Prin.  2.) 

4.  A  peach  orchard  contains  720  square  rods,  and  its  length  is  to 
its  breadth  as  5  to  4  ;  what  are  its  dimensions  ? 

Operation. — The  area  of  a  rectangle  5  by  4  equals  20  (898). 
20  :  720  : :  52  :  aj2  rr:  900  ;     ^900  =  30  rd.,  length. 
20  :  720  : :  42  :  a;2  =  576  ;     ^/^=  24  rd.,  width. 

5.  It  is  required  to  lay  out  283  A.  107  P.  of  land  in  the  form  of 
a  rectangle,  so  that  the  length  shall  be  3  times  the  width.  Find 
the  dimensions. 

6.  A  pipe  1.5  in.  in  diameter  fills  a  cistern  in  5  hours  ;  find  the 
diameter  of  a  pipe  that  will  fill  the  same  cistern  in  55  min.  6  sec. 

7.  The  area  of  a  triangle  is  24276  sq.  ft.,  and  its  sides  in  proportion 
to  the  numbers  13,  14,  and  15.     Find  the  length  of  its  sides  in  feet 


202  MENSURATION. 

8.  If  it  cost  $167.70  to  enclose  a  circular  pond  containing  17  A. 
110  P.,  what  will  it  cost  to  enclose  another  i  as  large  ? 

9.  If  63.39  rods  of  fence  will  enclose  a  circular  field  containing 
2  acres,  what  length  will  enclose  8  acres  in  circular  form  ? 

REVIEW    OF    PLANE    FIGURES. 

l*IiO  B  LJE  JI  S  , 

911.  1.  How  much  less  will  the  fencing  of  20  acres  cost  in  the 
square  form  than  in  the  form  of  a  rectangle  whose  breadth  is  J  the 
length,  the  price  being  $2.40  per  rod  ? 

2.  A  house  that  is  50  feet  long  and  40  feet  wide  has  a  square  or 
pyramidal  roof,  whose  height  is  15  ft.  Find  the  length  of  a  rafter 
reaching  from  a  corner  of  the  building  to  the  vertex  of  the  roof. 

3.  Find  the  diameter  of  a  circular  island  containing  IJ  sq.  miles. 

4.  What  is  the  value  of  a  farm,  at  $75  an  acre,  its  form  being  a 
quadrilateral,  with  two  of  its  opposite  sides  parallel,  one  40  ch. 
and  the  other  22  ch.  long,  and  the  perpendicular  distance  between 
them  25  chains  ? 

5.  Find  the  cost,  at  18  cents  a  square  foot,  of  paving  a  space  in 
the  form  of  a  rhombus,  the  sides  of  which  are  15  feet,  and  a  per- 
pendicular drawn  from  one  oblique  angle  will  meet  the  opposite 
side  9  feet  from  the  adjacent  angle. 

6.  A  goat  is  fastened  to  the  top  of  a  post  4  ft.  high  by  a  rope  50  ft. 
long.     Find  the  area  of  the  greatest  circle  over  which  he  can  graze. 

7.  How  much  larger  is  a  square  circumscribing  a  circle  40  rods 
in  diameter,  than  a  square  inscribed  in  the  same  circle  ? 

8.  What  is  the  value  of  a  piece  of  land  in  the  form  of  a  triangle, 
whose  sides  are  40,  48,  and  54  rods,  respectively,  at  the  rate  of 
$125  an  acre  ? 

9 .  The  radius  of  a  circle  is  5  feet ;  find  the  diameter  of  another 
circle  containing  4  times  the  area  of  the  first. 

10.  How  many  acres  in  a  semi-circular  farm,  whose  radius  is 
100  rods  ? 

11.  What  must  be  the  width  of  a  walk  extending  around  a  gar- 
den 100  feet  square,  to  occupy  one-half  the  ground? 

12.  An  irregular  piece  of  land,  containing  540  A.  36  P.  is  ex- 
changed for  a  square  piece  of  the  same  area  ;  find  the  length  of  one 
of  its  sides  ?  If  divided  into  42  equal  squares,  what  is  the  length 
of  the  side  of  each  ? 


SOLIDS. 


303 


13.  A  field  containing  15  A.  is  30  rd.  wide,  and  is  a  plane  inclining 
in  the  direction  of  its  length,  one  end  being  120  ft.  higher  than  the 
other.     Find  how  many  acres  of  horizontal  surface  it  contains. 

14.  If  a  pipe  3  inches  in  diameter  discharges  12  hogsheads  of 
water  in  a  certain  time,  what  must  be  the  diameter  of  a  pipe  which 
will  discharge  48  hogsheads  in  the  same  time  ? 

SOLIDS. 

912 .    A  Solid  or  Sody  has  three  dimensions,  length,  breadth, 
and  thickness. 
The  planes  which  bouud  it  are  called  its  faces^  and  their  intersections,  its 


913.  A  Prism  is  a  solid  whose  ends  are  equal  and  parallel, 
similar  polygons,  and  its  sides  parallelograms. 

Prisms  take  their  names  from  the  form  of  their  bases,  as  triangular^  quad- 
rangular^  pentagonal^  etc. 

914.  The  Altitude  oi  a  prism  is  the 
perpendicular  distance  between  its  bases. 

915.  A    Parallelopipedtni    is    a 

prism  bounded  by  six  parallelograms,  the 
opposite  ones  being  parallel 

916.  A   Cube  is  a  parallelopipedon 
whose  faces  are  all  equal  squares. 

917.  A  Cf/llnder  is  a  body  bounded 
by  a  uniformly   curved  surface,  its'  ends  being  equal  and  parallel 
circles. 

1.  A  cylinder  is  conceived  to  be  generated  by  the  revolution  of  a  rectangle 
about  one  of  its  sides  as  an  axis. 

2.  The  line  joining  the  centers  of  the  bases,  or  ends,  of  the  cylinder  is  its  aUi' 
tude^  or  axis. 


Cube. 


Triangular 
Prism. 


Quad  ran  2:ular 
Prism. 


Pentagonal 
Prism. 


Cylinder. 


204 


MENSURATION. 


I^ltOBLEMS. 

918.  To  find  the  convex  surlace  of  a  prism  or 
cylinder. 

1.  Find  the  area  of  the  convex  sur- 
face of  a  prism  whose  altitude  is  7  ft., 
and  its  base  a  pentagon,  each  side  of 
whicli  is  4  feet. 

Operation. — 4  ft.  x  5  =  20  ft.,  pert- 
metcT. 
20  ft.  X  7=140  sq.  ft.,  convex  surface. 

2.  Find  the  area  of  the  convex  sur- 
face of  a  triangular  prism,  whose  alti- 
tude is  8  J  feet,  and  the  sides  of  its  base 
4,  5,  and  6  feet,  respectively. 

Operation.  —4  ft.  +  5  f t.  +  6  ft.  = 
15  ft.,  perimeter. 
15  ft.  X  8J=127J  sq.  ft.,  contex surface. 

3.  Find  the  area  of  the  convex  surf qice  pf  a  cylinder  whose  altitude 
is  2  ft.  5  in.  and  the  circumference  of 
its  base  4  ftr^  jn. 

Operation.— 2  ft.  5  in. = 29  m. ;  4  ft. 
9  in,  =  57  in, 

57  in.  X  29  =  1653  sq.  in.  =  11  sq.  ft. 
?!^^        69  sq.  inches,  convex  surface. 

Rule. — Multiply  the  perimeter  of  the  hose  l)y  the  altitude. 
To  find  the  entire  surface,  add  the  area  of  the  bases  or  ends. 

4.  If  a  gate  8  ft.  high  and  6  ft.  wide  revolves  upon  a  point  In  its 
center,  what  is  the  entire  surface  of  the  cylinder  described  by  it  ? 

5.  Find  the  superficial  contents,  or  entire  surface  of  a  parallelo- 
pipedon  8  ft.  9  in.  long,  4  ft.  8  in,  wide,  and  3  ft.  3  in.  high. 

6.  What  is  the  entire  surface  of  a  cylinder  formed  by  the  revo- 
lution  about  one  of  its  sides  of  a  rectangle  that  is  6  ft.  6  in.  long 
and  4  ft.  wide  ? 

7.  Find  the  entire  surface  of  a  prism  whose  base  is  an  equilateral 
triangle,  the  perimeter  being  18  ft.,  and  the  altitude  15  ft. 


PYRAMIDS     AKD     COJSTES, 


205 


919.  To  find  the  volume  of  any  prism  or  cylinder. 

1.  Find  the  volume  of  a  triangular  prism,  whose  altitude  is  20  ft., 
and  each  side  of  the  base  4  feet. 

Opekation.— The  area  of  the  base  is  6.928  sq.  ft.  (882> 
6.928  sq.  ft.  x  20  =  138.56  cu.  ft.,  mlume. 

2.  Find  the  volume  of  a  cylinder  whose  altitude  is  8  ft.  6  in.,  an< 
the  diameter  of  its  base  3  feet. 

Operation.— 32  x  .7854  =  7.0686  square  feet,  area  of  base  (905). 
7.0686  sq.  ft.  x  8.5  =  60.083  cubic  feet,  wlume. 
Rule. — Multiply  the  area  of  the  base  by  the  altitude, 

3.  Find  the  solid  contents  of  a  cube  whose  edges  are  6  ft.  6  in. 

4.  Find  the  cost  of  a  piece  of  timber  18  in.  square  and  40  ft.  long, 
at  $.30  a  cubic  foot, 

5.  Required  the  solid  contents  of  a  cylinder  whose  altitude  is 
15  ft.  and  its  radius  1  ft.  3  in. 

6.  What  is  the  value  of  a  log  24  ft.  long,  of  the  average  circum- 
ference of  7.9  ft.,  at  $.45  a  cubic  foot  ? 


PYRAMIDS    AND    CONES.       . 

920.  A  JPyramid  is  a  body  having  for  its  base  a  polygon, 
and  for  its  other  faces  three  or  more  triangles,  which  terminate  in 
a  common  point  called  the  mrtex. 

Pyramids,  like  prisms,  take  their  names  from  their  baees,  and  are  called  tri' 
angular^  square^  or  quadrangular^  pentagonal^  etc. 


Frustum. 


Cone. 


Frustum. 


Pyramid. 

921.  A  Cone  is  a  body  having  a  circular  base,  and  whose  con- 
vex surface  tapers  uniformly  to  the  tertex. 

It  is  a  body  conceived  to  be  formed  by  the  revolution  of  a  right-angled  triangle 
about  one  of  its  sides  containing  the  right  angle,  as  an  immovable  axis. 

922.  The  Altitude  of  a  pyramid  or  of  a  cone  is  the  perpendic- 
ular distance  from  its  vertex  to  the  plane  of  its  base. 


306  MENSURATION. 

923.  The  Slant  Height  of  2i pyramid  is  the  perpendicular  dis. 
tance  from  its  vertex  to  one  of  the  sides  of  the  base  ;  of  a  cone,  is  a 
straight  line  from  the  vertex  to  the  circumference  of  the  base. 

924.  The  Frustum  of  a  pyramid  or  cone  is  that  part  which 
remains  after  cutting  off  the  top  by  a  plane  parallel  to  the  base. 

l^JiOBljJEMS. 

925.  To  find  the  convex  surface  of  a  pyramid  or 
cone. 

1.  Find  the  convex  surface  of  a  triangular  pyramid,  the  slant 
height  being  16  ft.,  and  each  side  of  the  base  5  feet. 

Operation.— (5  ft.  +  5  ft.  +  5  ft.)  x  16^2  =  120  sq.  ft.,  com.  surf. 

2.  Find  the  convex/Surface  of  a  cone  whose  diameter  is  17  ft.  6  in., 
and  the  slant  height  30  feet. 

Rule. — Multiply  the  perimeter  or  circumference  of  the  base  hy  one- 
half  the  slant  height. 
To  find  the  entire  surface,  add  to  this  product  the  area  of  the  base. 

3.  Find  the  entire  surface  of  a  square  pyramid  whose  base  is  8  ft. 
6  in.  square,  and  its  slant  height  21  feet. 

4.  Find  the  entire  surface  of  a  cone  the  diameter  of  whose  base 
is  6  ft.  9  in.  and  the  slant  height  45  ft. 

5.  Find  the  cost  of  painting  a  church  spire,  at  $.25  a  sq.  yd.,  whose 
base  is  a  hexagon  5  ft.  on  each  side,  and  the  slant  height  60  feet. 

926.  To  find  the  volume  of  a  pyramid  or  of  a  cone. 

1.  What  is  the  volume,  or  solid  contents,  of  a  square  pyramid 
whose  base  is  6  feet  on  each  side,  and  its  altitude  12  feet. 

Opekation.— 6  X  6  X 12 -r- 3  —  144  cu.  ft.,  wlume. 

2.  Find  the  volume  of  a  cone,  the  diameter  of  whose  base  is  5  ft. 
and  its  altitude  lOJ  feet. 

Operation.— (52  ft.  x  .7854)  x  lOJ-f-3  =  68.721^  cu.  ft.,  'oolume. 
Rule. — Multiply  the  area  of  the  lose  hy  one-third  the  altitude. 

3.  Find  the  solid  contents  of  a  cone  whose  altitude  is  24  ft.,  and 
the  diameter  of  its  base  30  inches. 

4.  What  is  the  cost  of  a  triangular  pyramid  of  marble,  whose 
altitude  is  9  ft.,  each  side  of  the  base  being  3  ft.,  at  $2^  per  cu.  foot  ? 

5.  Find  the  volume  and  the  entire  surface  of  a  pyramid  whose 
base  is  a  rectangle  80  feet  by  60  feet,  and  the  edges  which  meet  at 
the  vertex  are  130  feet. 


PYRAMIDS     AND     CONES.  207 

927.  To  find  the  convex  surface  of  a  frustum  of  a 
pyramid  or  of  a  cone. 

1.  What  is  the  convex  surface  of  a  frustum  of  a  square  pyramid, 
whose  slant  height  is  7  feet,  each  side  of  the  greater  base  4  feet,  and 
of  the  less  base  18  inches? 

Operation. — The  perimeter  of  the  greater  base  is  16  ft.,  of  the  less 

6  feet. 

16  ft. +  6  ft.  X  7-5-2  =  77  sq.  ft.,  co7ivex  surface. 

3.  Find  the  convex  surface  of  a  frustum  of  a  cone  whose  slant 
height  is  15  feet,  the  circumference  of  the  lower  base  BO  feet,  and 
of  the  upper  base  16  feet. 

Rule. — Multiply  the  sum  of  the  perimeters,  or  of  the  circumfer- 
ences, by  one-half  the  slant  height. 
To  find  the  entire  surface,  add  to  this  prodnct  the  area  of  both  ends,  or  bases. 

8.  How  many  square  yards  in  the  convex  surface  of  a  frustum 
of  a  pyramid,  whose  bases  are  heptagons,  each  side  of  the  lower 
base  being  8  feet,  and  of  the  upper  base  4  feet,  and  the  slant  height 
55  feet? 

928.  To  find  the  volume  of  a  frustum  of  a  pyramid 
or  cone. 

1.  Find  the  volume  of  the  frustum  of  a  square  pyramid  whose 
altitude  is  10  feet,  each  side  of  the  lower  base  12  feet,  and  of  the 
upper  base  9  feet. 


Operation.— 12' +  9'  =  225 ;  (225+  ^144x81)  x  10-^3=1110  cu. 
feet,  volume. 

2.  How  many  cubic  feet  in  the  frustum  of  a  cone  whose  altitude 
is  6  ft.  and  the  diameters  of  its  bases  4  ft.  and  3  feet  ? 

Rule. — To  the  sum  of  the  areas  of  loth  bases  add  the  square  root 
of  the  product,  and  multiply  this  sum  by  one-third  of  the  altitude, 

3.  How  many  cubic  feet  in  a  piece  of  timber  30  ft.  long,  the 
greater  end  being  15  inches  square,  and  that  of  the  less  12  inches  ? 

4.  How  many  cubic  feet  in  the  mast  of  a  ship,  its  height  being 
50  ft.,  the  circumference  at  one  end  5  feet  and  at  the  other  8  feet  If 


208  MElf  SURATION. 

THE    SPHERE. 

929.  A  Sphere  is  a  body  bounded  by  a  uniformly  curved  sur- 
face, all  the  points  of  which  are  equally  distant 
from  a  point  within  called  the  center. 

930.  The  Diameter  of  a  sphere  is  a 
straight  line  passing  through  the  center  of  the 
sphere,  and  terminated  at  both  ends  by  its 
surface. 

931.  The  Radius  of  a  sphere  is  a  straight  line  drawn  from 
the  center  to  any  point  in  the  surface. 

932.  To  find  the  surface  of  a  sphere. 

1.  Find  the  surface  of  a  sphere  whose  diameter  is  9  in. 
Operation.— 9  in.  x  3.1416  =  28.2744  in.,  circumference, 

28.2744  in.  X  9  =  254.4696  sq.  in.,  surface. 
Rule. — Multiply  the  diameter  hy  the  circumference  of  a  great  circle 
of  the  sphere. 

2.  What  is  the  surface  of  a  globe  3  feet  in  diameter  ? 

3.  Find  the  surface  of  a  globe  whose  riadiua  is  1  foot. 

933.  To  find  the  volume  of  a  sphere. 

1.  Find  the  volume  of  a  sphere  whose  diameter  is  18  inches. 
Operation. — 18  in.  x  3.1416  =  56.5488  in.,  circumference. 

56.5488  in.  x  18  =  1017.8784  sq.  in.,  mrface, 
1017.8784  sq.  in.  x  18-^6=3053.6352  cu.  in.,  mlume, 
'Rule.— Multiply  the  surface  by  \  of  the  diameter,  or  \  of  the  radius. 

2.  Find  the  volume  of  a  globe  whose  diameter  is  30  in. 

3.  Find  the  solid  contents  of  a  globe  whose  radius  is  5  yards. 

934.  To  find  the  three  dimensions  of  a  rectangu- 
lar solid,  the  volume  and  the  ratio  of  the  dimensions 
being  given. 

1.  What  are  the  dimensions  of  a  rectangular  solid,,  whose  volume 
is  4480  cu.  ft.,  and  its  dimensions  are  to  each  other  as  2,  5,  and  7  ? 


Operation.— <v/4480  ^  (2  x  5  x  7)  =  4 ;  4  ft.  x  2  =  8  ft.,  height . 
i  ft.  X  5  =  20  ft.,  yyidth;  4  ft.  x  7  =  28  ft.,  length. 


REVIEW     OF     SOLIDS.  209 

RuLE.—I.  Divide  the  wlume  hy  the  product  of  the  terms  proportional 
to  the  three  dimensions,  and  extract  the  cube  root  of  the  quotient. 

II.  Multiply  the  root  thus  obtained  by  each  proportional  term  ;  the 
products  will  be  the  corresponding  sides. 

2.  What  are  the  dimensions  of  a  rectangular  box  whose  volume 
is  3000  cu.  ft.,  and  its  dimensions  are  to  each  other  as  2,  3,  and  4  ? 

3.  A  pile  of  bricks  in  the  fonii  of  a  parallelopiped  contains  30720 
cu.  feet,  and  the  length,  breadth,  and  height  are  to  each  other  as  3, 
4,  and  5.     What  are  the  dimensions  of  the  pile  ? 

SIMILAR    SOLIDS. 

935.  Similar  Solids  are  such  as  have  the  same  form,  and 
diifer  from  each  other  only  in  volume. 

Principles. — 1.  The  volumes  of  similar  solids  a/re  to  each  other  as 
the  cubes  of  their  like  dimensions. 

1.  If  the  volume  of  a  cube  3  inches  on  each  side  is  27  cu.  in., 
what  is  the  volume  of  one  7  inches  on  each  side. 

Operation.— 33 :  7^ :  :  27  cu.  in. :  a;  =  343  cu.  in.,  volume. 

2.  The  like  dimensions  of  similar  solids  are  to  each  other  as  the  cube 
roots  of  their  volumes. 

3.  If  the  diameter  of  a  ball  whose  volume  is  27  cu.  in.  is  3  in., 
what  is  the  volume  of  one  7  inches  on  each  side  ? 

Operation.— ^27  :  -^343  ::  3  :  aj  =  7  in.  diameter. 

REVIEW    OP     SOLIDS. 
mOBLEMS, 

936.  1.  What  is  the  edge  of  a  cube  whose  entire  surface  is 
1050  sq.  feet,  and  what  is  its  volume  ? 

2.  What  must  be  the  inner  edge  of  a  cubical  bin  to  hold  1250  bu. 
of  wheat  ? 

3.  How  many  gallons  will  a  cistern  hold,  whose  depth  is  7  ft., 
the  bottom  being  a  circle  7  feet  in  diameter  and  the  top  5  feet  in 
diameter  ? 

4.  What  is  the  value  of  a  stick  of  timber  24  ft.  long,  the  larger 
end  being  15  in.  square,  and  the  less  6  in.,  at  28  cents  a  cubic  foot  ? 


310  MENSURATION. 

5.  If  a  cubic  foot  of  iron  were  formed  into  a  bar  ^  an  inch  square, 
without  waste,  what  would  be  its  length  ? 

6.  If  a  marble  column  10  in.  in  diameter  contains  27  cu.  ft.,  what 
is  the  diameter  of  a  column  of  equal  length  that  contains  81  cu.  ft.? 

7.  How  many  board  feet  in  a  post  11  ft.  long,  9  in.  square  at  the 
bottom,  and  4  in.  square  at  the  top  ? 

8.  The  surface  of  a  sphere  is  the  same  as  that  of  a  cube,  the  edge 
of  which  is  12  in.     Find  the  volume  of  each. 

9.  A  ball  4.5  in.  in  diameter  weighs  18  oz.  Avoir. ;  what  is  the 
weight  of  another  ball  of  the  same  density,  that  is  9  in.  in  diameter  ? 

10.  In  what  time  will  a  pipe  supplying  6  gal.  of  water  a  minute 
fill  a  tank  in  the  form  of  a  hemisphere,  that  is  10  ft.  in  diameter? 

11.  The  diameter  of  a  cistern  is  8  feet ;  what  must  be  its  depth 
to  contain  75  hhd.  of  water? 

12.  How  many  bushels  in  a  heap  of  grain  in  the  form  of  a  cone, 
whose  base  is  8  ft.  in  diameter  and  altitude  4  feet  ? 

GAUGING. 

937 .  Gauging  is  the  process  of  finding  the  capacity  or  volume 
of  casks  and  other  vessels. 

A  cask  is  equivalent  to  a  cylinder  having  the  same 
length  and  a  diameter  equal  to  the  mean  diameter  oi  the 
cask. 


To  find  the  mean  diameter  of  a  cask  {yearly). 
Add  to  the  head  diameter  f ,  or,  if  the  staves 
are  hut  little  curved,  .6,  of  the  difference  between  the  head  and  hung 
diameters. 

To  find  the  volume  of  a  cask  in  gallons. 

Multiply  the  square  of  the  mean  diameter  hy  the  length  {both  in 
inches)  and  this  product  hy  .0034. 

1.  How  many  gallons  in  a  cask  whose  head  diameter  is  24  inches, 
bung  diameter  30  in.,  and  its  length  34  inches  ? 

Operation. — 24  +  (30  —  24  x  |)  =r  28  in.,  mean  diameter, 
28^  X  34  X  .0034  -  90.63  gal.,  capacity. 

2.  What  is  the  volume  of  a  cask  whose  length  is  40  inches,  the 
diameters  21  and  30  in.,. respectively. 

3.  How  many  gallons  in  a  cask  of  slight  curvature,  3  ft.  6  in.  long, 
the  head  diameter  being  26  in.,  the  bung  diameter  31  in.  ? 


FORMULAS. 


811 


938. 


1.  The  Diameter  s 


The    Circum- 
ference 


3.  The  Area 


939. 


1.  The  Surface 


2.  The  Volume 


3 


CIRCLES. 

y  =  the  circumference, 

c  =  the  side  of  an  equal  square. 

\   =  the  side  of  an  inscribed  equi- 
\  lateral  triangle. 

^  =  the  sideof  an  inscribed  square, 
j-  =  the  diameter. 

j-  =  the  side  of  an  equal  square. 

\   =  the  side  of  an  inscribed  equi- 
\  lateral  triangle. 

\  —  the  side  of  an  inscribed  square, 

\  —  the  radius. 

—  the  square  of  the  radius, 
\  -  the  square  of  the  diameter. 

i    !07958  [  ~  the  sq're  of  the  circumference. 

SPHERES. 

{Circumference  x  itsdiam, 
Radius^  x  12.5664. 
IHametef^  x  3.1416. 
Circumference^  x  .3183. 
(Surface  x  J  its  diameter, 
Badius^  x  4.1888. 
Diameter^  x  .5286. 
Circumference^  x  .0169. 


X   3.1416 
-f-     .3183 
X      .8862 
-i-  1.1284 
X     .8660 
-f-     .1547 
X     .7070 
-^  1.4142 
X     .3183 
-i-  3.1416 
X     .2821 
-f-  3.5450 
X     .2756 
-^  3.6276 
X     .2251 
-i-  4.4428 
X     .15915 
-f-  6.28318 
r  -^  3.1416 
X   1.2732 
■i-     .7854 


The  Diatneter 

4.  The  Circumference 

5.  The  Madius 

6.  The  Side  of  Inscribed  Cube  ■• 


I  \^0f  surface  x  .5642. 
^ Of  volume  X  1.2407. 
I  ^Ofsu^if^e  X  1.77255. 

^ Of  volume  x  3.8978. 
j  ^J  Of  surface  x  .2821. 
\  ^  Of  volume  X  .6204. 
j  Radius  x  1.1547. 
\  Diameter  x  .5774. 


213 


BEVIEW. 


940. 


SYNOPSIS    FOR    EEVIEW. 


r  1.  Definition.     3.  Lines.     3.  Angles.     4.  Plane  Figubes. 


1.  Defs. 


o 

H 

12; 


5.    Tri- 
angles. 


2.  Prob- 
lems. 


Triangle.  2.  Right-angled  Tri.  3. . 
4.  ^056.  5.  Perpendicular.  6.  Altitude,  7.  Equi- 
lateral Triangle.  8.  Isosceles  Triangle.  9.  Scalene 
Triangle.  10.  Equiangular  Triangle.  11.  ^cw^e- 
angled  Triangle.    12.  Obtuse-angled  Triangle, 

f  Area  of  Triangle. 
Either  Dimension. 
Area  of  a  Triangle. 
The  Hypothenuse. 
The  Base  or  Perp. 


882. 
883. 
884. 
886. 
887. 


To  find 


y  Rule. 


,  Quad- 
rilat- 
erals. 


7.  Circle. 


1.  Defs. 


2.  Prob 
lems. 


I" 

(  898.) 
(  900.  ) 


Quadrilateral.  2.  Parallelogram.  3.  Rectangle, 
4.  Square.  5.  Rhornboid.  6.  PJioml)us.  7.  Trape- 
zoid,   8.  Trapezium.    9.  Altitude, 

Parallelogram.  \ 
Trapezoid.         >  Rule. 
Trapezium.        ) 


To  find 
area  of 


1.  Defs.       1.  Circle.       8.  Diameter,       3.  Radius. 


2.  Prob- 
lems. 


904. 
905. 
906. 
907. 
908. 
909. 


■To  find 


Diam.  or  Circum.    Rule,  1,  2. 
^/•ea.    Rule,  1,  2. 
Diam.  or  Circ.    Rule,  1,  2,  3. 
Side  of  Ins.  Square.    Rule,  1,  2. 
J.rea  q/"  Circular  Ring.    Rule. 
^  JlfeaTi  Proportional.    Rule. 


Similar  Plane  Figures.       1.  Defs.       2.  Prin.  1,  2,  3,  4,  5. 


9.  SOUDS.  - 


1.  Defs. 


.  Prob- 
lems. 


(  1.  Solid  or  Body.  2.  Prism.  3.  Altitude.  4.  Par- 
alldopipedon.  5.  Cwde.  6.  Cylinder.  7.  Pyra- 
mid. 8.  Cone.  9.  Altitude  of  Pyramid  or  Cone. 
10.  Slant  Height.  11.  Frustum.  12. 
13.  Diameter.    14.  Radius. 


918. 

919. 

925. 

926. 

927.  }^  To  find 

928. 

932. 

933. 

934.  J 


3.  Similar  Solids. 
10.  GA.UGING.       1.  Definitions. 


Conv.Surf.  of  Prism  or  Cyl.  Rule. 
Volume         "  "       Rule.^ 

Com.Surf.ofPyr.  or  Cone.  Rule. 
Volume  "  "         Rule. 

Cbzzv.  Surf,  of  Frustum.     Rule. 
Fo^^^me         "  "  Rule. 

Surface  of  Sphere.  Rule. 

Fo/Mm6  "        "  Rule. 

^  Dim.  of  Rectang.  Solid.      Rule. 

1    Defs.       2.  Principles^  1,  2. 

2.  Rules. 


< 


The  edges  of  this  cube  are  each  1  Me^ter^  or  10  Dec/i-me^ters,  or  100  Cen^ti- 

me'ters^  m  length. 


ScAiiE,  ^  of  the  Exact  Size* 

94:1.  The  MetTic  System  of  weights  and  measures  is  based 
upon  the  decimal  notation,  and  is  so  called  because  its  primary  unit 
is  the  Metier. 

942.  The  Me'ter  (m.)  is  the  base  of  the  system,  and  is  the 
one  ten -millionth  part  of  the  distance  on  the  earth's  surface  from  the 
equator  to  either  pole,  or  39.37079  inches. 

Me^ter  means  measure ;  and  the  three  principal  units  are  units  of  lengthy 
capacity  or  wlume^  and  weight. 


21'i  METRIC     SYSTEM. 

943.  The  Multiple  UnifSf  or  higher  denominations,  are 
named  by  prefixing  to  the  name  of  the  primary  units  the  Greek 
numerals,  Dek'a  (10),  Hek'to  (100),  KU'o  (1000),  and  Myr'ia  (10000). 

Thus,  1  dek'a-me'ter  (Dm.)  denotes  10  me'ters  (m.) ;  1  hek'to-me'ter  (Hm.)^ 
100  me'ters ;  1  kil'o-me'ter  (Km.),  1000  me'ters ;  and  1  myr'ia-me'ter  (Jfm.), 
10000  meters. 

944:.  The  Snb-fnultiple  Units^  or  lower  denominations, 
are  named  by  prefixing  to  the  names  of  the  primary  units  the  Latin 
ordinals,  Dec'i  (y^),  Cen'ti  (j^J^),  Mil'li  (ywff)- 

Thus,  1  dec'i-me'ter  (dm.)  denotes  ^,  or  .1  of  a  me'ter ;  1  cen'ti-me'ter  (cm.), 
tJs,  or  .01  of  a  me'ter;  1  milli-me'ter  {mm\  t^i  or  .001  of  a  me^ter. 

Hence,  it  is  apparent  from  the  ruwie  of  a  unit  whether  it  is  greater  ot  less  than 
the  standard  unit,  and  also  how  m^my  Um£S. 

945.  The  Metric  System  being  based  upon  the  decimal  scale y  the 
denominations  correspond  to  the  orders  of  the  Arabic  Notation  ;  and 
hence  are  written  like  United  States  Money,  the  lowest  denomina- 
tion at  the  right.     Thus, 


g 

Ob 

^ 

s 

0) 

o 

1-i 

OQ 

3 

'2 

CD 

J 

0 

E-< 

H 

» 

^ 

6 

7 

0 

1 

1 

«« 

i 

i 

^ 

^ 

ft 

fii 

5       '2 


P    f^    Eh       w       h 


*« 


§ 


I 


The  number  is  read,  67015.638  me'ters.  It  may  be  expressed  in 
other  denominations  by  placing  the  decimal  point  at  the  right  of  the 
required  denomination,  and  writing  the  name  or  abbreviation  after 
the  figures. 

Thus,  the  above  may  be  read,  670.15638Hm. ;  or  67.015638  Km. ; 
or  670156.38  dm. ;  or  6701563.8  cm. ;  or  it  may  be  read, 

6  Mm.  7  Km.  0  Hm.  1  Dm.  5  m.  6  dm.  3  cm.  8  mm. 

Write  3672.045  me'ters^  and  read  it  in  the  several  orders  ;  read  it 
in  kil'o-me'ters ;  in  hek'to-me'ters ;  in  dek'a-me'ters ;  in  dec'i- 
me'ters ;  in  cen'ti-me'ters. 

The  names  miU,  cent^  dim£,  used  in  United  States  Money,  correspond  to 
mil^li,  cent%  de&U  in  the  Metric  Systeim  Hence  the  eagle  might  be  called  the 
dek'a-doUar^  since  it  is  10  dollars ;  the  dime,  a  decfirdoUar,  eince  it  is  xV  of  a 
dollar,  etc. 


METRIC     SYSTEM. 


215 


MEASURES    OP    LENGTH. 

946.  The  Mefter  is  the  unit  of  length,  and  is  equal  to  89.37  in. 
or,  1.0936  yd.  +. 


Metric  Denominations.  U.  S.  Value. 

1  Mil'li-me'ter  =  .08937  in. 
10  Mil'li-me'ters,  mm.  =  1  Cen'ti-me'ter  —  .8937  in. 
10  Cen'ti-me'ters,  cm.  =  1  Dec'i-me'ter  =  3.937  in. 
10  Dec'i-me'ters,   dm.  =  1  Meter  =     39.37  in. 

10  Me'ters,  m,    =1  Dek'a-me'ter  =  32.809  ft. 

10  Dek'a-me'ters,  Dm.  =  1  Hek'to-me'ter=:19.8842  rd. 
10  Hek'to-me'ters,J3m.  =  1  Kiro-me'ter  =  .6213  mi. 
10  Kil'o-me'ters,   Km.  =  1  Myr'ia-me'ter=:  6.2138  mi. 

Units  of  long  measure  form  a  scale  of  tens; 
hence,  in  writing  numbers  expressing  length,  one 
decimal  place  must  be  allowed  for  each  denomina- 
tion. 

Thus,  9652  mm.  may  be  written  965.2  cm.,  or 
96.52  dm.,  or  9.652  m ,  or  .9652  Dm. 

1.  The  Metier  is  used  in  measuring  cloths  and  short  dis- 
tances. 

2.  The  KU'o-me'ier  is  commonly  used  for  measuring  long 
distances,  and  is  about  |  of  a  common  mile. 

3.  The  Cent'i-me'ter  and  MU'li-ine'ter  are  used  by  mechanics 
and  others  for  minute  lengths. 

4.  In  business,  Dgc'i-me'^ers  are  usually  expressed  in  CenVi- 
me'ters. 

5.  The  BeWa-me'ter,  Bek'to-rm'ter.  and  Myr'ia-me'ter  are 
seldom  used,  but  their  values  are  expressed  as  EM'o-me'ters. 


EXERCISES. 


Read  the  following : 


8.9  m.  . 

36  dm. 
428  cm. 
6.57  dm. 


346  Dm. 

57.9  Hm. 
479.6  m. 
36.75  mm. 


451  Hm. 
593.7  Km. 
105.6  Dm. 
6000  Km. 


4  in,     1  ffm 


ii: 


II 

cokl 

.  ? 


13.043  Km. 
500.032  m. 
31045.7  cm. 


216  METRIC     SYSTEM. 

Change  the  following  to  metiers : 

327  Dm.  947  cm.  0.72  Km.  30674  mm. 

28  Hm.  236  dm.  1.73  Hm.  83.062  cm. 

16.8  Km.  43.5  cm.  35.4  Dm.  4000.5  dm. 

1.  Write  6  kilometers  6  dekameters  6  meters  6  decimeters  6  centi- 
meters.    Ans.  6.06666  Km.,  or  m.^m^  Hm.,  or  606.666  Dm.,  etc. 

Write  the  following,  expressing  each  in  three  denominations]: 

2.  24379  dm.;     15032036  cm.:    2475064  mm.;    30471  Dm. 

3.  6704  Hm. ;    85  Km.  ;     120000  m. ;    780109  cm.  ;  75  m. 

Similar  examples  should  be  given,  until  the  pupil  is  familiar  with  the  reduc- 
tion of  higher  to  lower,  and  of  lower  to  higher  denominations,  by  changing  the 
place  of  the  decimal  point  and  using  the  proper  abbreviations. 

947.  To  add,  subtract,  multiply,  and  divide 
Metric  Denoniiuations. 

1.  What  is  the  sum  of  314.217  m.,  53.062  Hm.,  and  225  cm.  ? 
Operation.    314.217  m.  +  5306.2  m.  +  2.25  m.  =  5622  667  m.,  Ans. 

2.  Find  the  difference  between  4.37  Km.  and  1246  m. 
Opbration.    4.37  Km.  —  1.242  Km.  =  3.128  Km.,  Ans. 

3.  How  much  cloth  in  8 J  pieces,  each  containing  43.65  m.  ? 
Operation.    43.65  m.  x  8.25  =  384.8625  m.,  Am. 

4.  How  many  garments,  each  containing  3.5  m.,  can  be  made 
from  a  piece  of  cloth  containing  43.75  Dm.  ? 

Operation.    437.5  m.  -s-  3.5  m.  =  125  times;  hence,  125  garments,  Ans. 

Rule. — Reduce  the  given  numbers  to  the  same  denominations, 
when  necessary  ;  then  ^proceed  as  in  the  corresponding  operations  with 
whole  numbers  and  decimals. 

EXEItCISES, 

1.  Add  7.6  m.,  36.07  m.,  125.8  m.,  and  9.127  m. 

2.  Express  as  meters  and  add  475  dm.,  3241  cm.,  and  725  mm. 

3.  Add  56.07  m.,  1058.2  dm.,  430765  cm.,  6034.58  m.,  and  express 
the  result  in  kilometers. 

4.  From  8.125  Km.  take  3276.4  m.  Ans.  4.8486  Km. 


METRIC     SYSa:EM.  217 

5.  The  distance  around  a  certain  square  is  3.15  Krn.     How  many 
meters  will  a  man  travel  who  walks  around  it  4  times? 

6.  How  many  meters  of  ribbon  will  be  required  to  make  32  badges, 
each  containing  40  centimeters  ?  Ana.  12.8  m. 

7.  What  will  be  its  cost,  at  15  cents  a  meter? 

8.  Find  the  difference  between  25.3  Km.  and  425.25  m. 

9.  If  an  engine  runs  36.8  Km.  in  an  hour,  how  far  does  it  run 
between  8  o'clock  and  12  o'clock  ? 

10.  In  what  time  will  a  train  fun  from  Boston  to  Albany,  at  the 
rate  of  46.55  Km,  per  hour,  the  distance  being  about  325.85  Km.  ? 

11.  From  a  piece  of  cloth  containing  45.75  m.,  a  tailor  cut  5  suits, 
each  containing  7.5  m.     How  much  remained  ? 

12.  A  wheel  is  3.6  m.  around.     How  many  times  will  it  revolve  in 
rolling  a  distance  of  1.08  Km.?  Arts,  300. 

MEASURES    OF    SURFACE. 

948.  The  units  of  square  measure  are 
squares,  the  sides  of  which  are  equal  to  a  unit 
of  long  measure.  1  sq.  cm.,  Exact  Size, 


lOOSq.Decl-me'ters  ^W^''^''      ^    i  =\f^ 

( 1  Centar  {ca.)    )       (1.1 

lOOSq.Me'ters  = -} !  1' ^f"*.         Ui' 


100  Sq.  Mirii-me'ters(«g.?wm.)  =     1  sq.  cm,  =  0.155  sq.  in. 

100  Sq.  Cen'ti-me'ters  =     1  sq.  dm.  =    15.5  sq.  in. 

[10.764  sq.ft. 
.196sq.yd. 
\  3.954  sq.  rd. 
.0247  acre. 

100  Sq.  Dek^a-me'ters  =  \]  l^'  f/^' . „   x  !■  =  2.471  acres. 

( 1  Hektar  (Ea.) ) 

100  Sq.  Hek'to-me'ters  =     1  sq.  Km.  =  .3861  sq.  mi. 

Units  of  square  measure  form  a  scale  of  hundreds;  hence,  in 
writing  numbers  expressing  surface,  two  decimal  places  must  be 
allowed  for  each  denomination. 

Thus,  36  sq.  m.  4  sq.  dm.  27  sq.  cm.  are  written  36.0427  sq.  m.  ; 
and  6  Ha.  5  a.  ^  ca.  are  written  6.0503  Ha.,  or  605.03  a.,  etc. 

1.  The  Square  Me'teris  the  unit  for  measuring  ordinary  surfaces  of  small 
extent,  as  floors,  ceilings,  etc. 

2.  The  Ar,  or  Square  Bek'a-me'ter,  is  the  unit  of  land  measure,  and  is  equal 
to  119.6  sq.  yd.,  or  3  954  sq.  rd.,  or  .0247  acre. 


218  METRIC     SYSTEM. 


EXEMCTS  ES. 


1.  Read  36145  sq.  m.,  naming  each  denomination. 

Ans.  3  sq.  Hm.  61  sq.  Dm.  45  sq.  m. 

2.  Write  in  one  number  4  of  each  denomination  from  sq.  Hm.  to 
sq.  mm.,  expressed  in  sq.  Hm.  Ans.  4.0404040404  sq.  Hm. 

3.  Express  the  following,  each  in  three  denominations ; 
6  sq.  Km.  6  sq.  Hm.  24  sq.  Dm.  5  sq.  m. ; 

16  sq.  Dm.  8  sq.  m.  4  sq.  dm.  15  sq.  cm. 

4.  In  15  sq.  Hm.  how  many  square  meters? 

5.  What  is  the  surface  of  a  floor  12  m.  long  and  7  m.  wide  ? 

6.  Add  8  times  4  Ha.,  7  times  9  a.,  and  12  times  14  ca. 

7.  What  is  the  area  of  a  piece  of  land  42  Dm.  long  and  36  Dm. 
wide?  Ans.  1512  sq.  Dm.,  or  15.12  Ha. 

8.  Divide  125000  ca.  into  8  equal  parts. 

9.  How  many  times  is  2.50  sq.  m.  contained  in  5  Ha.  ? 

10.  How  many  meters  of  carpeting  0.6  m.  wide  will  cover  a  floor 
8  m.  long  and  5.7  m.  wide?  Ans.  76  m. 

11.  At  15  cents  a  sq.  m.,  what  is  the  cost  of  painting  a  surface 
20.5  m.  long  and  6.8  m.  wide?  Ans.  $20.91. 

12.  A  man  having  5  Ha  8  a.  7  ca.  of  land,  sold  .3  of  it,  at  $25  an 
ar.    What  did  he  receive  for  what  he  sold  ? 


MEASURES    OP    VOLUME. 

949.  The  units  of  cubic  measure  are  cubes, 

the  edfices  of  which  are  equal  to  a  unit  of  long  ^  _ 

^  ^  °   leu.  cm.,  Exact  Size. 

measure. 


1000  Cu.  MiVli-meHers  {cu.  mm.)  =     1  cu.  cm.       =      .061  cu.  in. 

.^^^  ^     r.     ,..       ,.  \1  cu.  dm.   )       S  .0353  < 

1000  Cu.  Cen'ti-me'ters  =  -j  ^  ^.,^^^^^^^  f  ^^-j  ^^^.^^ 

....  ^     T.    ,.       ,.  \\eu.7n.  35.316 

1000  Cu.  Dec'i-me'ters  ==  -j  ^  g^^^  ^^^  J  --j     ^^^ 


L.0567  li.  qt. 

;.3165cu.ft. 

759  cord. 

Units  of  cubic  measure  form  a  scale  of  thousands;  hence,  in 
writing  numbers  expressing  volume,  three  decimal  places  must  be 
allowed  for  each  denomination. 

Thus,  42  cu.  m.  31  cu.  dm.  5  cu.  cm.  are  written  42.031005  cu.  m. 
The  cubic  dec^lmeUer,  wheu  used  as  a  unit  of  liquid  or  dry  measure,  is  called 
a  Wter. 


METRIC     SYSTEM.  219 


WOOD    MEASURE.. 


1000  Cu.  Dec'i-me^ters  {cu,  dm.)  )  _  \1  cu.  m.    |  _  j  .2759  cord. 
10  Dec^-sters  {ds.)  )  ~"  \\Ster,8.  \  "  (85.3165  cu.  ft. 

10  Sters  =  1  Dek'a-ster,  Da,    =    2.759  cord. 


Units  of  wood  measure  form  a  scale  of  tens  ;  hence,  but  one  deci- 
mal is  required  for  each  denomination. 

Thus,  9  Ds.  4  s.  7  ds.  are  written  94.7  s.  ;    or  9.47  Ds. 

1.  The  Cubic  Metier  is  the  unit  for  measuring  ordinary  solids ;  as  excavations, 
embankments,  etc. 

2.  Cubic   Cen'ti-me'ters  and  MiVli-me'ters  are  used  for  measuring  minute 
bodies. 

3.  The  CvMc  Me'ter  when  used  as  a  unit  of  measure  for  wood  or  stone  is 
called  a  8ter. 

4.  The  common  Cord  is  about  the  same  as  3.6  sters ^  or  36  de&i-sters. 

EXERCISES, 

1.  Write  30  Ds.  6  s.  8  ds.  Ans.  30.68  Ds. 

2.  Express  in  cu.  m.,  3  cu.  m.  3  cu.  dm.  3  cu.  cm.  3  cu.  mm. 

Ans.  3.003003008  cu.  m. 

3.  Write  and  read  the  following,  each  in  cu.  dm.,  in  cu.  cm.,  and 
in  cu.  mm.  : 

16  cu.  m.   275  cu.  dm.  ;     204  cu.  m.   .016  cu.  dm.   .024  cu.  cm.  ; 
10  cu.  m.    324  cu.  dm.    .016  cu.  cm.    3244  cu.  cm. 

4.  Express  in  cu. meters  and  add  :  7  cu.m.,  55  cu.  dm.,  12  cu.  m., 
6  cu.  dm.,  15  cu.  cm.,  10532  cu.  cm.  Ans.  19.071547  m. 

5.  From  36  cu.  m.  subtract  8  times  42  cu.  dm.    Ans.  35.664  m. 

6.  How  many  cubic  meters  of  brick  in  a  wall  16  m.  long,  3  m. 
high,  and  8  dm.  thick?  Ans.  38.4  cu.  m. 

7.  How  many  cu.  meters  of  earth  niust  be  removed  in  digging  a 
cellar  16.5  m.  long,  8.2  m.  wide,  and  3.2  m.  deep? 

8.  In  a  pile  of  wood  9.3  m.  long,  2.8  m.  high,  and  1.5  m.  wide, 
how  many  sters  ?  Ans.  39.06  s. 

9.  At  $2.25  a  ster,  what  would  be  the  cost  of  a  pile  of  wood  5.6  m. 
long,  3.4  m.  wide,  and  2.5  m.  high  ? 

10.  If  a  cu.  centimeter  of  silver  is  worth  $.75,  what  is  the  value 
of  a  brick  of  silver  12.4  cm.  long,  3.6  cm.  wide,  and  2.5  cm.  thick? 


220 


METRIC     SYSTEM. 


MEASURES    OF    CAPACITY. 

950.  The  Li'ter  is  the  unit  of  ca- 
pacity, both  of  Liquid  and  of  Dry 
Measures,  and  is  equal  in  volume  to  one 
cu.  dtci-me'ter,  equal  to  1.0567  qt.Liquid 
Measure,  or  .908  qt.  Dry  Measure. 


lOMirii-li^ters,  m?.=l  Cen'ti-li'ter  - 

10  Cen^ti-U'ters,  c?.  =1  Dec'i-li'ter  - 

10  Dec'i-li'ters,   dl.  -1  Ijiter 
IOLi'terb,  /.  =1  Dek'a-li^ter  - 

10  Dek'a-li'ters,  J)l  =  \  Hek'to-li^ter 

10  HekHo-li'ters,5?.=l  Kil'o-li'teror  Stem 

10  Ril'o-li'ters,    Kl.=\  Myr'ia-li'ter (Ml.)^ 


Dry  M, 

.61  cu.  in. 
:   6.10   "     " 
.908  qt. 
r     9.081  *'     : 
:     2.837  bu.  : 
(28.37bu. ) 
'jl.BOScu.ydJ 
:283.73bu.    : 


Liquid  M. 

=.338fl'doz. 
=    .845  gi. 
=1.0567  qt. 
=2.64175  gal. 
=26.4175  " 

=264.175  " 

=2641.75  " 


1.  The  Li'ter  is  used  in  measuring  liquids  in  moderate  quantities. 

2.  The  Hek'to-Wter  is  used  for  measuring  grain,  fruit,  roots,  etc.,  in  large 
quantities,  also  wine  in  casks. 

3.  Instead  of  the  KiVo-Wter  and  MiVli-me'ter^  the  Cubic  Me'ter  and  Cuby: 
Cen'ti-me'tery  which  are  their  equals,  may  be  used. 


ext:ti  c isljs, 

1.  Write  5  kiloliters  5  liters  5  deciliters  5  centiliters. 

Ans.  5.00555  Kl.,  or  5005.551. 

2.  Read,  naming  each  denomination,  the  following  : 

45624  cl.  ;    306721  ml.  ;    76031  dl.  ;    89764  i. 

3.  In  3846  1.  how  many  cl.  ?    How  many  Dl.  ?    Kl.  ?    dl.  ?    ml.  ? 
'      4.  Find  the  sum  of  175  1.,  25  HI.,  42  cl.,  and  16  dl. 

5.  From  6  times  25  HI.  take  15  times  36  I. 

6.  Divide  5  HI.  of  corn  equally  among  25  persons.       Ans.  20 1. 

7.  From  a  cask  of  wine  containing  2  HI.  of  wine,  125  1.  were 
drawn  out.     How  much  remained  ? 

8.  How  many  HI.  of  wheat  can  be  put  into  a  bin  3  m.  long,  2  m. 
wide,  and  1.5  m.  deep?  Ans.  90  HI. 

9.  What  must  be  the  length  of  a  bin  1.5  m.  wide,  1  m.  deep,  to 
contain  7500  liters  of  grain  ?  Ans.  5  m. 


METRIC     SYSTEM.  221 


MEASURES    OF    WEIGHT. 

951.  The  Gram  is  the  unit  of  weighty  and  is  equal  to  the 

weight  of  a  cu,  ceii^ti-me' ter  of  distilled  water. 

A  Gram  is  equal  to  15.432  gr.  Troy,  or  .03527  oz.  Avoir. 

10  Mirii-grams,        mg,     =  1  Cen'ti-gram     =        .1543  +  gr.  Tr. 
10  Cen'ti-grams,       eg,      =  1  Dec' i -gram      =       1.5432+  **      " 

10  Dec'i-grams,         dg,      =  1  Gram  ^-l^^fflJ""    1 

\     .03527+ oz.Av. 

IOGkams,  g.        =  1  Dek'a-gram     -        .3527+"      " 

10  Dek'a-grams,       Dg,     =  1  Hek'to-gram   =      3.5274+  "      '* 

-I  A  XT  1  /+  Tj  -,  ( Kiro-gram, )         (  2.6792     lb.  Tr. 

10  Hek'to-grams,      Hg.     =  1<         ^?.„      >•=    i  ^  ^^.^     „     . 

^  ^  j    or  KWo   )         i  2.2046+  lb.  Av. 

10  Kiro-grams,         Kg.     =  1  Myr'ia-gram    ■=     22.046  + 

10  Myr'ia-grams,  IJg.,  or  \ 

100  Kil'os, 

lOQuin'tals,  Q.,  or 

1000  Kilos,  K  f  ~  "]       or  Ton  {~  I  1.1023  +    tons. 


•1  =  1  Quin'tal  =  220.46     + 

[  _  .  j  Tonneau,     |  __  j  22 
)  (       orTouf~U.: 


1.  The  Gram  is  used  for  weighing  letters,  gold,  silver,  medicines,  and  all 
small,  or  costly  articles. 

2.  The  KWo-gram  or  KiVo  is  the  weight  of  a  cu.  dm.  of  water,  and  is  the  unit 
of  common  weight  in  trade,  being  a  trifle  less  than  2|  lb.  Avoir. 

3.  The  Ton  is  the  weight  of  a  cu.  m.  of  water,  and  is  used  for  weighing  very- 
heavy  articles,  being  about  294^  lb.  more  than  a  common  ton. 

4.  The  Avoir,  oz.  is  about  28  g. ;  the  pound  is  a  little  less  than  ^  a  kilo. 


BXEBCISBS. 

1.  Read  340642  eg.  in  grams ;  in  hectograms;  in  kilograms. 

2.  Change  16.5  T.  to  kilos  ;  to  grams  ;  to  decigrams. 

3.  If  coffee  is  $.80  a  kilo,  what  will  5  quintals  cost  ? 

4.  How  many  boxes  containing  1  gram  each,  will  be  required  to 
liold  1  kilo  of  quinine  ?  Ans.  1000. 

5.  If  a  letter  weighs  3.5  g.,  how  many  such  letters  will  weigli 
1.015  Kg.?    -  Ans.  2^0. 

6.  A  car  weighing  6.577  T.  contains  125  barrels  of  salt,  each 
weighing  102.15  K.     What  is  the  weight  of  the  car  and  contents  ? 

7.  Find  the  difference  in  the  weight  of  the  car  and  its  contents  ? 


222  METKIC     SYSTEM. 

952»  To  change  the  Metric  to  the  Common  Sys- 
tem, 

1.  In  3.6  Km.,  how  many  feet? 

OPERATION.  Analysis.— The  meter  is 

3.6  Km.  X  1000  =  3600  m.  the  principal  unit  ofthe  table; 

,^^^^  ,  hence,  reduce  the  kilometers 

39.37  in.  x  3600  =  1417^2  in.  ^^  ^^^^^8.    Since  there  are 

141732  in.  -f-  12         =  11811  ft.,  An^,      39.37   inches   in   1  meter,  in 

3600  m.  there  are  3600  times 
39.37  in.,  or  141732  in.  =  11811  ft.    Therefore,  8.6  Km.  are  equal  to  11811  ft. 

Rule. — Beduce  the  metric  nnmber  to  the  denomination  of  the 
principal  unit  of  the  table ;  then  multiply  by  the  equivalent ,  and 
reduce  the  product  to  the  required  denomination. 


jjxERC  is:es. 

2-  How  many  feet  in  472  centimeters?  Ans.  15.485  ft 

3.  How  many  cubic  feet  in  2000  sters  ? 

4.  How  many  gallons,  liquid  measure,  in  325  deciliters  ? 

5.  How  many  gallons  in  108.24  liters?      Ans.  28  gal.  2.77  qt. 

6.  How  many  bushels  in  3262  kiloliters  ? 

7.  How  many  acres  in  436  ats?  Avs.  10.774  A. 

8.  In  942325  centiliters,  bow  many  bushels? 

9.  In  456  kilograms,  how  many  pounds  ?       Ans,  1005.024  lb. 

10.  In  42  ars,  bow  many  square  rods  ? 

11.  Change  75.5  hektars  to  acres.  Ans,  186.56  A. 

12.  How  many  gallons  in  24J  liters  of  wine  ? 

13.  How  many  pounds  of  butter  in  124  kilos? 

14.  In  28  sters,  how  many  cords?  An^.  7.725  C. 

15.  In  72  kilometers,  how  many  miles  ? 

16.  Change  148  grams  to  ounces  Avoirdupois.       Ans,  5.22  oz. 

17.  Change  150.75  kilos  to  pounds. 

18.  How  many  sq.  rods  in  5  a.  85  ea.  ?  Ans  23.13  sq,  rd. 

19.  What  is  the  weight  of  24  cu.  dm.  148  cu.  cm.  of  silver,  if  a 
cu.  centimeter  weighs  11.4  g.?  Ans.  737.553  lb.  Tr. 


METRIC     SYSTEM.  223 

953.  To  change  the  Common  to  the  Metric  Sys- 
tem. 

I.  In  10  lb.  4  oz.  Troy,  how  many  kilograms? 

OPERATION.  Analysis.— The  gram, 

10  lb.  4  oz.  —  10.25  lb.  the  principal  unit  of  the 

10.25  lb.   X  5760  =  59040  gr.  *^*^1^'    ^^   expressed     in 

erains  :  hence,  reduce  the 
59040 gr.-15.432gr.  =  3825.75  g.  %^,^  ,,^  '^^,,es    to 

3825.75  g.  -^  1000  ==  3.82575  Kg.,  Ans,      grains.     Since  15.433  gr. 

make  1  gram,  there  are 
as  many  grams  in  59040  gr.  as  15.432  gr.  is  contained  times  in  59040  gr.,  or 
3825.75  g.  And  since  there  are  1000  grams  in  a  kilogram,  dividing  3885.75  g.  hy 
1000  g.,  the  quotient  is  3.82575.    Therefore,  there  are  3.82575  Kg.  in  10  lb.  4  oz. 

Rule. — Reduce  the  given  quaTvtity  to  the  denomination  in  which 
the  equivalent  of  the  principal  unit  of  the  metric  table  is  expressed  ; 
divide  by  this  equivalent,  and  reduce  the  quotient  to  the  required 
denomination. 

EXEItCISES. 

2.  In  6172.9  lb  av.,  how  many  kilograms  ?   Ans.  2800.009  Kg. 

3.  How  many  ars  in  a  square  mile  ? 

4.  How  many  cu.  decimeters  in  1892  cu.  feet  ? 

5.  In  892  gr.,  how  many  grams?  Ans.  57.8  g. 

6.  In  2  mi.  272  rd.  5  yd.,  how  many  kilometers?  Ans.  4.59  Km. 

7.  How  many  sters  in  264.4  cu.  feet  ? 

8.  How  many  liters  in  3  bu.  1  pk.  ?  Ans.  114.5  1. 

9.  How  many  grams  in  6  lb.  Troy  ?    In  6  lb.  Avoir.  ? 
10.  How  many  meters  in  3  mi.  272  rd.  ? 

II.  In  1828  cu.  yd.  how  many  cu.  meters?  Ans.  1397.52  cu.  m. 

12.  In  3588  sq.  yards,  how  many  sq.  meters? 

13.  Bought  454  bu.  of  wheat,  at  $3  a  bushel,  and  sold  the  same 
at  18.75  per  hektoliter ;  how  many  hektoliters  did  I  sell  ?  Did  I 
gain  or  lose,  and  how  much  ?  Ans.  160  HI. ;  gain,  $38. 

14.  In  13  gal.  3  qt.  2  pt.  3  gi.,  how  many  liters? 

Ans.  53.351.+. 

15.  Sow  many  sq.  meters  of  plastering  in  a  room"  18  ft.  6  in. 
long,  14  ft.  wide,  and  9  ft.  6  in.  high?         Ans.  55.367  sq.  in.  +. 


224  METRIC     SYSTEM. 


TEST     PEOBLEMS. 

954.     1.  Find  the  weight  of  a  barrel  of  flour  (196  lb.)  in  Kg.  ? 

2.  What  is  the  cost  of  a  carpet  for  a  room  10.5  m.  long,  and  8.4  m. 
wide,  if  the  carpet  is  84  cm.  wide  and  costs  $2.75  a  meter? 

Ans.  $288.75. 

3.  A  farmer  sold  540  HI.  of  wheat,  at  $2  a  bushel,  and  invested 
the  proceeds  in  coal  at  $7  per  ton.     How  many  tons  did  he  buy  ? 

Ans.  437.785  T.  +  . 
4  What  is  the  cost  of  a  building  lot  75  m.  long  and  63  m.  wide, 
at  $40  an  ar  ?  Ans,  $1860. 

5.  A  bushel  of  wheat  weighs  60  lb.  What  is  the  weight  of  5  HI. 
of  wheat,  in  kilograms  ?  Ans.  386.05  Kg. 

6.  What  will  be  the  cost  of  a  pile  of  wood  15. 7  m.  long,  3  m. 
high,  and  7.53  m.  wide,  at  $1.50  a  ster? 

7.  The  new  silver  dollar  weighs  412 J  gr.  Troy.  How  many 
grams  does  it  weigh  ?  A7is.  26.73  g. 

8.  How  many  acres  of  land  in  24.6  Km.  of  a  highway,  which  is 
20  m.  wide  ?  Ans.  121.573  A. 

9.  A  bin  is  4.2  m.  long,  2.8  m.  wide,  and  1.5  m.  deep.  What  will 
be  the  cost  of  filling  it  with  charcoal,  at  25  cts.  a  hektoliter  ? 

10.  A  merchant  bought  300  m.  of  silk  in  Lyons,  at  12.5  francs  a 
meter  ;  he  paid  75  cents  a  yard  for  duty  and  freight,  and  sold  it  in 
New  York  at  $5  a  yard.     What  was  his  gain  ?         Ans.  $670.61. 

11.  What  price  per  pound  is  equivalent  to  $2.50  per  Hg.  ? 

13.  If  a  man  buys  5000  g.  of  jewels,  at  35  francs  a  gram,  and  sells 
them  at  $15  a  pennyweight,  what  was  his  gain  or  loss  ? 

13.  If  a  field  produces  40  HI.  of  oats  to  the  hektar,  how  many 
bushels  is  that  to  the  acre?  Ans.  45.93  bu. 

14.  What  price  per  peck  is  equivalent  to  80  cts.  a  dekaliter  ? 

15.  What  will  be  the  cost  of  excavating  a  cellar  18.3  m.  long, 
10.73  m.  wide,  and  3.4  m.  deep,  at  20  cents  per  ster  ? 

16.  How  many  pounds  Avoir,  are  there  in  96.4  kilos  of  salt  ? 

17.  How  many  liters  will  a  cistern  hold  that  measures  on  the 
inside  5.5  ft.  long,  4  ft.  6  in   wide,  and  4  ft.  djep  ?  Ans.  3803.383  I. 


METRIC     SYSTEM.  225 

18.  How  many  meters  of  lining  that  is  60  cm.  wide  will  line 
15  m.  of  silk  that  is  75  cm.  wide  ?  Ans.  18,75  cm. 

19.  A  lady  bought  40.5  m.  of  silk  in  Paris.  What  would  be  its 
value  in  Boston,  at  $4  75  per  yard  ? 

20.  A  bin  is  4  m.  long,  2.3  m.  wide.  How  deep  must  it  be  to 
contain  40  HI.  of  grain?  Ans.  4.347  +  dm. 

21.  How  many  sters  of  wood  can  be  piled  in  a  shed  8.5  m.  loag, 
5.8  m.  wide,  and  4.2  m.  high  ?  What  would  be  its  value  at  $3.25  a 
cord?  ^;is.  207.03  8.;  $185,665. 

22.  A  dray  is  loaded  with  60  bags  of  grain,  each  bag  holding 
8  Dl. ;  allowing  75  K.  of  grain  to  the  hectoliter,  what  is  the  weight 
of  the  load  in  metric  tons  ?  Ans.  3.6  T. 

23.  How  many  meters  of  shirting,  at  $.18  per  meter,  must  be 
given  in  exchange  for  250  HI.  of  oats,  at  $1.20  per  hectoliter? 

24.  A  merchant  shipped  to  France  50  barrels  of  sugar,  each  con- 
taining 250  lb.,  paying  $2  per  cwt.  for  transportation.  He  sold  the 
sugar  at  $.34  per  kilogram,  and  invested  the  proceeds  in  broadcloth, 
at  $4  per  meter.    How  many  yards  did  he  purchase  ? 

25.  A  cu.  decimeter  of  copper  weighs  8.8  Kg.  What  is  the  value 
of  a  bar  of  the  same  metal  15  dm.  long,  9.6  cm.  broad,  and  6.4  cm. 
thick,  at  $1.30  a  kilogram?  Ans.  $105.43. 

26.  How  many  bricks,  each  20  cm.  lon^  and  10  cm.  wide,  will 
pave  a  walk  95.4  m.  long  and  2.1  m.  wide;  and  what  will  they 
cost,  at  $1.75  per  hundred ?  Ans.  10017  bricks ;  $175,297. 

27.  What  is  the  value  of  a  pile  of  wood  40  ft.  6  in.  long,  4  ft. 
broad,  and  6  ft.  6  in.  high,  at  $6.50  per  dekastere  ? 

28.  What  will  be  the  cost  of  building  a  wall  96  Dm.  6  m.  8  dm. 
long,  1  m.  6  dm.  thick,  and  2  m.  4  cm.  high,  at  $6.75  a  cu.  metiV? 

29.  A  wine  merchant  imported  to  Boston  1000  dekaliters  of  wine, 
at  a  cost  of  $.75  a  liter,  delivered.  At  what  price  per  gallon  must 
he  sell  the  same  to  clear  $2000  on  the  shipment  ?       Ans.  $3,596. 

30.  How  many  gallons  of  water  will  a  cistern  contain  that  is  3  m. 
deep,  2  m.  long,  and  1.5  m.  wide;  and  what  will  be  its  weight  in 
metric  tons  ?  Ans.  2377.575  gals. ;  9  T. 


226 


METRIC     SYSTEM. 


TABLE    OF    EQUIVALENTS. 

955.  The  equivalents  here  given  agree  with  those  that  have 
been  established  by  Act  of  Congress  for  use  in  legal  proceedings  and 
in  the  interpretation  of  contracts. 


1  inch  =  2.540  centimeters. 
1  foot  =  3.048  decimeters. 
1  yard  =:  0.9144  meter. 
1  rod  —  0.5029  dekameter. 
1  mile  =  1.6093  kilometers. 
1  sq.  in.  =  6.452  sq.  centimeters. 
1  sq.  ft.  =:  9.2903  sq.  decimeters. 
1  sq.  yard  =  0.8361  sq.  meter. 
1  sq.  rd.  =  25.293  sq.  meters. 
1  acre  =  0.4047  hektar. 
1  sq.  mile  =:  2.590  sq.  kilometers. 
1  cu.  in.  =  16.387  cu.  centimeters. 
1  cu.  ft.  r=  28.317  cu.  decimeters. 
1  cu.  yard  =  0.7645  cu.  meter. 
1  cord  =  3.624  sters. 
1  liquid  quart  =  0.9463  liter. 
1  gallon  =  0.3785  dekaliter. 
1  4ry  quart  =  1.101  liters. 
1  peck  =  0.881  dekaliter. 
1  bushel  =  3.524  dekaliters. 
1  ounce  av.  =  28.35  grams. 
1  pound  av.  =  0.4536  kilogram. 
1  T.  (2000  lbs.)  =  0.9072  met.  ton. 
1  grain  Troy  =  0.0648  gram. 
1  ounce  Troy  =  31.1035  grams 
1  pound  Troy  =  0.3732  kilogram. 


1  centimeter  =  0.3937  inch. 
1  decimeter  =  0.328  foot. 
1  meter  =  1.0936  yds.  -  39.37  in. 
1  dekameter  =  1.9884  rods. 
1  kilometer  =  0.62137  mile. 
1  sq.  centimeter  =  0.1550  sq.  in. 
1  sq.  decimeter  =  0. 1076  sq.  ft. 
1  sq.  meter  —  1.196  sq.  yards. 
1  ar  =r  3.954  sq.  rods. 
1  hektar  =  2.471  acres. 
1  sq.  kilometer  =  0.3861  sq.  mi. 
1  cu.  centimeter  —  0.0610  cu.  in. 
1  cu.  decimeter  =  0.0353  cu.  ft. 
1  cu.  meter  =  1.308  cu.  yards. 
1  ster  =  0.2759  cord. 
1  liter  =  1.0567  liquid  quarts. 
1  dekaliter  =  2.6417  gallons. 
1  liter  =  0.908  dry  quart. 
1  dekaliter  =  1.135  pecks. 
1  hectoliter  =  2.8375  bushels. 
1  gram  =  0,03527  ounce  Av. 
1  kilogram  =  2.2046  pounds  Av. 
1  metric  ton  =:  1 .1023  tons. 
1  gram  =  15.432  grains  Troy. 
1  gram  =  0.03215  ounce  Troy. 
1  kilogram  =  2.679  pounds  Troy. 


PARTIAL    PAYMENTS.  227 

VERMONT   RULE   EOE  PARTIAL  PAYMENTS. 

956.  The  General  Statutes  of  Vermont  provide  the  following 
HuLE  for  computing  interest  on  notes,  when  partial  payments  have 
been  made : 

*'  On  all  notes y  hills,  or  other  similar  obligations,  whether  made 
payable  on  demand  or  at  a  specified  time,  with  interest,  when 
payments  are  made,  such  payments  shall  be  applied :  first,  to  liqui- 
date the  interest  that  has  accrued  at  the  time  of  such  payments ; 
and,  secondly,  to  the  extinguishment  of  the  principal. 

*'  On  all  notes,  bills,  or  other  similar  obligations,  whether  made 
payable  on  demand  or  at  a  specified  time,  with  interest  annu- 
ally, the  annual  interests  that  remain  unpaid  shall  be  subject  to 
simple  interest,  from  the  time  they  become  due  to  the  time  of  final 
settlement ;  but  if  in  any  year,  reckoning  from  the  time  such  annual 
interest  began  to  accrue,  payments  have  been  made,  the  amount  of 
such  payments  at  the  end  of  such  year,  with  interest  thereon  from  the 
date  of  payment,  shall  be  applied :  first,  to  liquidate  the  simple  inter- 
est that  has  accrued  upon  the  unpaid  annual  interests  ;  secondly,  to 
liquidate  the  annual  interests  that  have  become  due;  and  thirdly,  to 
the  extinguishment  of  the  principal.'* 

EXERCISES. 

$3458.  Bradford,  Vt.,  Sept.  13,  1869. 

1.  For  value  received,  I  promise  to  pay  E.  W.  Colby  or  order  three 
thousand  four  hundred  and  fifty-eight  dollars,  on  or  before  the  first 
day  of  January,  1878,  with  interest.  Samuel  S.  Green. 

Indorsed  as  follows:  Dec.  16,  1870,  $100;  May  1,  1871,  $1000; 
Jan.  13,  1874,  $85 ;  April  13, 1876,  $450.75. 

What  was  due  Jan.  1,  1878?  Ans.  $3239.90. 

%^'^^'  St.  Johnsbury,  Vt.,  Nov.  22,  1868. 

^  2.  For  value  received,  I  promise  to  pay  James  Ferguson  or  order 
eight  hundred  and  seventy  two  dollars,  on  demand,  with  interest 
annually.  Sylyanus  E.  Boyle. 

Indorsed  as  follows :  April  4,  1869,  $28 ;  July  10,  1872,  $94.40  • 
Dec.  10,  1874,  $6.72 ;  Jan.  14,  1877, 

What  was  due  Dec.  28,  1878  ? 


228  PARTIAL    PAYMENTS. 


OPERATION. 

Int.  oil  Yearly 

Int.  Int.        Prin, 

int.  of  prin.  to  Nov.  22,  1869  .     ,    .    .    .  $52.32 

Am't  of  1st  payment 29.06 

Bal.  of  unpaid  yearly  int.     ......  23.26 

Int.  of  prin.  to  Nov.  22,  1872 156.96 

Int.  on  1  year's  int.  3. years $9.42 

int.  on  bal.  of  unpaid  yearly  int.  3  years  .  4.1J^       13.61 

193^83 
Am't  of  2d  payment 96.48 

Bal.  of  unpaid  yearly  int 97.35 

Int.  of  prin.  to  Nov.  22,  1875 156.96 

Int.  on  1  year's  int.  3  years 9.42 

Int.  on  bal.  of  unpaid  yearly  int.  3  years   .17.52 

26.94    25431 
Am't  of  3d  payment 7.10 

Bal.  of  int.  on  int 19.84 

Int  of  prin.  to  Nov.  22,  1877 104.64 

Int.  on  1  year's  int  1  year 3.14 

Int.  on  bal.  of  unpaid  yearly  int.  2  years  .  30.52       53.50      412.45 

1284.45 
Am't  of  4tli  payment 416.33 

New  principal 868.12 

Int.  of  new  prin.  to  Dec.  28,  1878 57.30 

Int.  on  1  year's  int.  1  mo.  6  d .31 

Due,  Dec.  28,  1878 $925.73 

ExPL\NATiON.— We  compute  the  interest  for  one  year  from  the  date  of  the 
note,  as  a  payment  is  made  within  that  year,  and  deduct  the  amount  of  the  pay- 
ment at  the  end  of  the  year  from  the  interest  due.  The  balance  of  interest  hears 
interest  till  Nov.  22, 1872.  The  amount  of  the  payment  at  the  end  of  this  year 
exceeds  the  interest  on  interest  due.  We  therefore  deduct  the  amount  of  the 
payment  from  the  total  interest  due,  and  have  a  balance  of  unpaid  yearly  inter- 
est, $97.35,  which  bears  simple  interest  till  Nov.  22,  1875.  At  this  date  the 
amount  of  the  payment  is  less  than  the  interest  on  interest  due.  We  there- 
fore deduct  the  amount  of  the  payment  from  the  amount  of  interest  on  interest, 
and  have  a  remainder  of  $19.84,  which  is  without  interest.  The  amount  of  un- 
paid yearly  interest  at  this  date  bears  simple  interest  till  the  next  balance. 


PARTIAL    PAYMENTS.  220 

The  amount  of  the  fourth  payment,  Nov.  22, 1877,  exceeds  the  total  interest 
due.  We  therefore  deduct  it  from  the  sum  of  the  interest  and  principal.  The 
remainder  forms  a  new  principal,  which  bears  simple  interest  to  the  settlement 
of  the  note,  Dec.  28, 1878,  and  one  year's  interest  on  the  same  bears  interest  from 
Nov.  22, 1878,  to  Dec.  28,  1878,  which  interest,  added  to  the  new  principal,  gives 
the  amount  due  Dec.  28,  1818— $925.73. 

In  cases  of  annual  interest  with  partial  payments,  like  the  above 
example,  obssrve  the  following  notes  ; 

1.  To  avoid  compounding  interest,  keep  the  principal,  unpaid  yearly  inter- 
ests, and  interest  on  yearly  interet^t,  in  separate  columns. 

2.  Deduct  the  amount  of  the  payment  or  payments  at  the  end  of  the  year 
from  the  interest  on  the  unpaid  yearly  interest,  when  it  does  not  exceed  this 
interest.  The  remainder  never  draws  interest,  but  is  liquidated  by  the  first  pay- 
ment that  equals  or  exceeds  it. 

3.  Deduct  the  amount  of  the  payment  or  payments  at  the  end  of  the  year 
from  the  sum  of  the  unpaid  yearly  interests  and  the  interest  on  the  unpaid 
yearly  interests,  when  this  amount  exceeds  the  interest  on  the  interest,  but  is 
less  than  such  sum.  The  remainder  is  a  balance  of  unpaid  yearly  interest  which 
draws  simple  interest  imtil  canceled  by  a  payment. 

4.  Deduct  the  amount  of  the  payment  or  payments  at  the  end  of  the  year 
from  the  sum  of  the  total  interest  due  and  the  principal,  when  it  exceeds  the 
total  interest  due.  The  remainder  forms  a  new  principal,  with  which  proceed 
as  with  the  original  principal. 

$5000.  Newport,  Vt.,  Oct.  19,  1862. 

3.  For  "oalue  received,  we  jointly  and  severally  promise  to  pay  John 
Smith  or  hearer  five  thousand  dollars,  sixteen  years  after  date,  with 
interest  annually,  Geo.  S.  Leazer. 

E.  D.  Crawford. 

Indorsed  as  follows :  Jan.  13,  1866,  |393 ;  Sept.  24,  1866,  $48 ; 
July  10,  1869,  $493.47;  Oct.  14,  1873,  $100;  Dec.  12,  1877,  $3200; 
April  15, 1878,  $65. 

What  was  due  Oct.  19,  1878?    Ans,  $7056.17. 


Burlington,  Yt.,  March  23,  1872. 

^.  For  value  received,  I  promise  to  pay  Jas.  B.  Vinton  or  order 
four  hundred  and  tweiity  dollars,  six  years  from  date,  with  interest 
annually.  Geo.  A.  Bancroft. 

Indorsed  as  follows ;  Oct.  3,  1873,  $40.23 ;  March  1,  1874,  $8 : 
Sept.  13, 1875,  $33.38. 

What  was  due  March  23, 1878  ?    Ans.  $494.62. 


230 


PARTIAL    PAYMENTS. 


Barton,  Vt.  Aug.  20,  1873. 
5.  For  value  receivedy  I  promise  to  pay  E.  J.  Baxter  or  order  six 
hundred  and  thirty-nine  dolla/rSj  on  demand,  with  interest  annually, 

Samuel  Macomber. 
Indorsed  as  follows  :  Oct.  14,  1877,  $10  ;  Dae.  24,  1878,  $20. 
What  was  due  March  30,  1879  ?    Ans,  $904.58.    " 


TABLE. 

Showing  amount  of  $1.00  from  1  to  20  years,  at  ^  5,  6,  7  and  8  per 
cent.y  Annual  Interests 


Years. 

4  per  cent. 

5  per  cent. 

6  per  cent, 

7  per  cent. 

8  per  cent. 

Years. 

1  . 

$1  0400 

$1.0500 

$1.0600 

$1.0700 

$1.0800 

.   1 

2  . 

1.0816 

1.1025 

1.1236 

1.1449 

1.1664 

.  2 

3  . 

1.1248 

1.1575 

1.1908 

1.2247 

1.2592 

.  3 

4  . 

1.1696 

1.2150 

1.2816 

1.3094 

1,3584 

.  4 

5  . 

1.2160 

1.2750 

1.3360 

1.3990 

1.4640 

.  5 

6  . 

1.2640 

1.3375 

14140 

1.4935 

1.5760 

.  6 

7  . 

1.3136 

1.4025 

1.4956 

1.5929 

1.6944 

.  7 

8  . 

1.3648 

1.4700 

1.5808 

1.6972 

1.8192 

.  8 

9  . 

1.4176 

1.5400 

1.6696 

1.8064 

1.9504 

.  9 

10  . 

1.4720 

1.6125 

1.7620 

1.9205 

2.0880 

.  10 

11  . 

1.5?80 

1.6875 

1.8580 

2.0395 

2.2320 

.  11 

12  . 

1.5856 

1.7650 

1.9576 

2.1634 

2.3824 

.  12 

13  . 

1.6448 

1.8450 

2.0608 

2.2922 

2  5392 

.  13 

14  . 

1.7056 

1.9275 

2.1676 

2.4259 

2.7024 

.  14 

15  . 

1.7680 

1.0125 

2.2780 

2.5645 

2.8720 

.  15 

16  . 

1.8320 

2.1000 

2.3920 

2.7080 

3.0480 

.  16 

17  . 

1.8976 

2.1900 

2.5096 

2.8564 

3.2304 

.  17 

18  . 

1.9648 

2.2825 

2.630S 

3.0097 

3.4192 

.  18 

19  . 

2.0336 

2.3775 

2.7556 

3.1679 

3.6144 

.  19 

20  . 

2.1040 

2.4750 

2.8840 

3.3100 

3.8160 

20 

ASSESSMEiq^T    OF   TAXES.  231 


VEEMONT  METHOD   OP  ASSESSING  TAXES. 

957.  The  Grand  List  is  the  base  on  which  all  taxes  are  assessed  ; 
it  is  Ifo  of  the  appraised  value  of  the  real  estate  and  personal 
property,  together  with  the  poll  list. 

The  Poll  List  is  $2.00  for  every  male  inhabitant,  from  21  to  70 
years  of  age,  except  such  as  are  specially  exempt  by  law. 

The  General  Statutes  of  Vermont  provide  that  the  listers  in  each 
town  shall  make  a  list  of  all  the  real  estate  and  personal  property, 
and  the  number  of  taxable  polls  in  such  town,  and  that  the  said 
list  shall  contain  the  following  particulars  : 

*'  First.  The  name  of  each  taxable  person. 

*'  Second,  The  number  of  polls  and  the  amount  at  which  the  same  are  set  in 
the  list. 

*'  TMrd,  The  quantity  of  real  estate  owned  or  occupied  by  such  person. 

"  Fourth,  The  value  of  such  real  estate. 

"  Fifth.  In  the  fifth  column  the  full  value  of  all  taxable  personal  estate  owned 
by  such  person. 

"  Sixth.  In  the  sixth  column  shall  be  set  the  one  per  centum  on  the  value  of 
all  personal  and  real  estate,  tojjether  with  the  amount  of  the  polls,  which  sum 
shall  be  the  amount  on  which  all  taxes  shall  be  made  or  assessed. 

The  State  and  County  Taxes  are  assessed  by  the  Legislature. 

The  minimum  of  the  State  School  and  Highway  Taxes  is  fixed  by 
law,  and  a  higher  rate  left  optional  with  the  town. 

A  Town  Tax  is  assessed  by  vote  of  the  town,  a  Village  Tax  by 
vote  of  the  village,  and  a  School  District  Tax  by  vote  of  the  district, 

EXERCISES. 

1.  The  town  of  Montpelier  voted  a  town  tax  of  $2.60  on  each 
dollar  of  the  grand  list.  The  appraised  value  of  the  real  estate  was 
$702727,  and  of  the  personal  property  $309987,  and  there  were 
740  taxable  polls.  What  was  the  grand  list  of  the  town?  How 
much  money  wfis  raised  by  this  vote  ?  Wh^t  was  John  Hammond's 
town  tax,  who  was  30  years  of  age,  and  whose  property  was  ap- 
praised at  $8927.75? 


232  ASSESSMENT    OF    TAXES. 

OPERATION. 

$702727  +  $309987= $1012714,  assessed  value  of  the  property. 
$1012714  X  .01  =$10127.14,  1  %  of  the  assessed  value. 
$2.00  X  740= $1480,  the  poll  list. 
$10127.14-4- $1480=111607.14,  the  grand  list. 
$2.60  X  11607.14=$30178.56,  amount  of  money  raised. 
$8927. 75 X. 01  =  $89.28,1%  of  the  assessed  value  of  John  Ham- 
mond's property. 

$89.28  +  $2.00,  his  poll  list  =  $91.28,  John  Hammond's  grand  list. 
$2.60  X  91 .28 =$237. 33,  John  Hammond's  town  tax. 

2.  The  appraised  value  of  property,  both  real  and  personal,  in 
the  town  of  Rutland,  for  the  year  1878,  was  $3415264.  The  num- 
ber of  taxable  polls  was  2066.  The  town  voted  to  raise  a  tax  of 
$28713.48.     What  was  the  tax  on  a  dollar  of  the  grand  list  ? 

Ans,  $0.75. 

3.  The  appraised  value  of  the  real  estate  in  the  city  of  Burling^ 
ton  was  $2542373;  of  the  personal  property,  $399937.  There 
were  2040  taxable  polls.  The  city  voted  to  raise  $60305.58  city 
tax.  What  was  the  amount  of  Henry  Cook's  tax,  a  resident,  who 
was  73  years  of  age,  and  whose  real  estate  was  appraised  at  $750, 
and  his  personal  property  at  $475.50  ?  Ans.  $22.06. 

4.  The  grand  list  in  the  town  of  Chelsea  was  $4403.74.  The  ap- 
praised valug  of  all  the  property  was  $368774.  How  many  taxable 
polls  were  there  in  that  town  ?  Ans.  358. 

5.  The  estimated  cost  of  schools  in  school  district  No.  8,  in  the 
town  of  Cabot,  for  one  year,  was  $765.  The  amount  of  public 
money  received  from  the  town  was  $71.50.  The  appraised  value  of 
the  real  estate  in  the  district  was  $48545  ;  of  the  personal  estate 
$15428.75 ;  the  number  of  taxable  polls  in  the  district  103.  How 
much  tax  on  a  dollar  of  the  grand  list  must  the  district  vote,  to  pay 
its  expenses  ?  Ans.  $0.82. 

6.  James  Bell  resides  in  Hardwick ;  he  is  44  years  of  ag-e ;  his 
property,  both  real  estate  and  personal,  is  appraised  at  $8975.50. 
Hardwick  voted  a  town  tax  of  $1.60  on  a  dollar  of  the  grand  list. 
The  highTvay  tax  is  $0.40 :  the  state  tax  is  $0.45  ;  the  state  school 
tax  is  $0.09  ;  the  school  tax  is  $0.86  ;  and  the  county  tax  $0.01,  on 
the  dollar.     What  is  the  amount  of  his  taxes  ?  Ans.  $315.64. 


.^x^IZg^^ag^Cv-.       g-^^^ 


MEASTJUE  S  E^ 


1.  A  Measure  is  a  standard  unity  established  by 
law  or  custom,  by  which  quantity,  as  extent,  dimension, 
capacity,  amount,  or  value  is  measured  or  estimated. 

Thus,  the  standard  unit  of  Measures  of  Extension  is  the  yard ; 
of  Liquid  Measure,  the  wine  gallon  ;  of  Dry  Measure,  the  Winches- 
ter hushel;  of  Weight,  the  Troy  'pound ,  etc.  Hence  the  length  of  a 
piece  of  cloth  is  ascertained  by  applying  the  yard  measure ;  the 
capacity  of  a  cask,  by  the  use  of  the  gallon  measure  ;  of  a  bin,  by 
the  use  of  the  hushel  measure ;  the  weight  of  a  body,  by  the  pound 
weight,  etc. 

3*  Measures  may  be  classified  into  six  hinds  : 

4.  Time. 


1.  Extension. 

2.  Capacity. 

3.  Weight. 


5.  Angles  or  Arcs. 

6.  Money  or  Value. 


MEASUEES   OF  EXTEITSIOK 

3.  Extension  has  length,  Ireadth,  and  thichness. 
4t  A  Line  has  length  only. 

5.  A  Surface  or  Area  has  length  and  breadth. 

6.  A  Solid  has  length,  Ireadth,  and  thichness. 


^34  EXTEiq^SIOiq'. 


LINEAR     MEASURE. 

'7.  Linear  Measure^  also  called  Long  MeaS' 
ure   IS  used  in  measuring  lines  and  distances. 


Table. 

13    Inches  (in.)  =  1  Foot  ....  /if. 

3    Feet  =  1  Yard  .    .    ,     .  pd. 

5i  Yards,  or)  ^  ^  ^^^  .    .    .    ,  rd. 
\U  Feet          ) 

320    Rods  -  1  Mile  ....  mi. 


1  ML  = 


63360  in, 

5280 /If. 

1760  yd. 

320  rd. 


8.  Cloth  Measure  is  practically  out  of  use.  In 
measuring  goods  sold  by  the  yard,  the  yard  is  divided 
into  halves,  fourths,  eighths,  and  sixteenths, 

2i  Inches  =  1  Sixteenth,  -jV  yd. 

2  Sixteenths,  (4^  in.)  =  1  Eighth.  ^  yd. 
2  Eighths,  (9  in.)  =  1  Quarter,  \  yd. 
4    Quarters  =  1  Yard,  1  yd. 

At  U.  S.  Custom-Houses,  in  estimating  duties,  the  yard  is  divided 
into  tenths  and  hundredtJis. 

9.  Mariners  use  the  following  denominations  : 

9    Inches  =  1  Span,  Sp. 

8    Spans  or  6  Ft.  =  1  Fathom,  fath. 

120    Fathoms  =  1  Cable's  Length,  c.  I. 

74  C.  Length  =  1  Nautical  Mile  (or  Knot),  mi. 

3  Miles  =  1  League,  lea. 

10.  In  geographical  and  astronomical  calculations  : 

1  Geographic  Mile  =  1.152|  Statute  Miles. 

3  "  *'  =  1  League. 

60  "  "  or )  _  ^  j^       j  of  Latitude  on  a  Meridian, 

69.16  Statute         "       )  (or  of  Long,  on  the  Equator 

360  Degrees  =  the  Circumference  of  the  Earth. 


MEASURES     OF     EXTENSION. 


235 


11.  The  following  are  sometimes  used  : 

3  Barley-corns,  or  Sizes  =  1  Inch.      Used  by  slioemakers. 

4  Inches  =  1  Hand. 


,  J     j  to  measure  the  height  of 
(   horses  at  the  shoulder. 


8-1%  Feet 
3  Inches 
21.888  Inches 
6  Points 
12  Lines 


=  1  Pace. 

=  1  Palm. 

=z  1  Sacred  Cubit. 

=  1  Line.  ) 

=  1  Inch. ) 


Used  in  clock-making. 


1.  The  nautical  mile  (or  knot)  is  the  same  as  the  geographical 
mile,  and  is  used  in  measuring  the  speed  of  vessels. 

2.  The  geographical  mile  is  (3V  of  3^77  or  j-rhoo  of  the  distance 
round  the  center  of  the  earth.  It  is  a  small  fraction  more  than  1.15 
statute  miles. 

3.  The  length  of  a  degree  of  latitude  varies,  being  68.72  miles  at 
the  equator,  68.9  to  69.05  miles  in  middle  latitudes,  and  69.30  to 
69.34  miles  in  the  polar  regions.  The  mean  or  average  length, 
69.16,  is  the  standard  recently  adopted  by  the  U.  S.  Coast  Survey. 
A  degree  of  longitude  is  greatest  at  the  equator,  where  it  is  69.16 
miles,  and  it  gradually  decreases  toward  the  poles,  where  it  is  0. 

12.  Surveyors^  Linear  Measure  is  used  by 
land  suryeyors  in  measuring  roads  and  boundaries  of 
land. 

Table. 

7.92  Inches  r==  1  Link     .  . 

25  Links    =  1  Rod      .  . 

4  Rods      =  1  Chain  .  . 

80  Chains  =  1  Mile     .  . 

1.  A  Gtmter^s  Chain  is  the  unit  of  measure,  and  is  4  rods,  or 
66  feet  long,  and  consists  of  100  links. 

2.  Engineers  commonly  use  a  chain  or  measuring  tape,  100  feet 
long,  each  foot  divided  into  tenths. 

3.  Measurements  are  recorded  in  chains  and  hundredths. 


I 

'  63330  in. 

rd. 
ch. 

1  Mi.  =  ^ 

8000  I. 
320  rd. 

mi. 

'&<Sch. 

236 


MEASURES     OF     EXTEKSION. 


13. 


COMPARISON    OF    DISTANCES. 


Country. 

Distance.      U. 

S.mile. 

Country. 

Distance. 

U.  S.  mile. 

England, 

1  Mile        = 

1 

Russia, 

1  Verst 

=     .66 

France, 

1  Km.        = 

.62 

Turkey, 

1  Berri 

=  1.04 

Spain, 

1  League  = 

4.15 

Portugal, 

IMilha 

=  1.28 

Prussia, 

1  Meile      = 

4.93 

Persia, 

1  Farsang 

=  4.17 

Austria, 

1  Meile      = 

4.98 

China, 

ILi 

=     .35 

Sweden, 

IMil          = 

QM 

Egypt, 

1  Mill 

=  1.15 

Switzerland, 

1  Lieue      = 

2.98 

East  Indies, 

1  Coss 

=  1.14 

Mexico, 

1  SiUo       = 

6.76 

Japan. 

IRi 

=2.562 

SURFACE     OR   SQUARE    MEASURE. 

14.  Surface  or  Square  Measure  is  used  in 
computing  areas  or  surfaces ;  as  of  land,  boards,  paint- 
ing, plastering,  paving,  etc. 


Table. 


144    Square  Inches  (Si 
9    Square  Feet 
30  J:  Square  Yards 
160    Square  Rods 
640    Acres 

sq.  mi.  A.        sq,  rd. 


sq.ft. 


,  in.)  =  1  Square  Foot 

=  1  Square  Yard   .     .  sq.  yd. 

=  1  Sq.  Rod  or  Perch  sq.  rd.;  P. 

=  1  Acre A. 

=  1  Square  Mile    .     .  sq.  mi. 

sq.  yd.  sq  ft.  sq.  in. 

1  =  640  =  102400  =  3097600  =  27878400  =  4014489600 
1  =        160  =^       4840  =       43560  =^        6272640 
1  =  30i=  272J:=  39204 

1  =  9  =  1296 

1  =  144 

15.  Artificers  estimate  their  work  as  follows  : 
By  the  square  foot ;  as  in  glazing,  stone-cutting,  etc. 
By  the  square  yard,  or  by  the  square  of  100  square  feet ;  as  in 
plastering,  flooring,  roofing,  paving,  etc. 

One  thousand  shingles,  averaging  4  in.  wide,  and  laid  5  in.  to  the 
weather,  are  estimated  to  be  a  squa/re. 


MEASURES     OF     EXTEKSIOK.  237 

16.  Surveyors^  Square  Measure  is  used  by 
surveyors  in  computing  the  area  or  contents  of  land. 

Table. 


625  Square  Links  {sq,  I.)  =1  Pole P. 

sq.  ch. 


16  Poles  =  1  Square  Chain 

10  Square  Chains  =  1  Acre     .     .     . 


.  A. 

.  sq.  mi, 

.  Tp. 

l. 


640  Acres  =  1  Square  Mile  . 

36  Square  Miles  (6  miles  square)  =  1  Township     . 

Tp.  sq.  mi.      A.  sq.  ch.  P.  sq. 

1  =  36  =  23040  =  230400  =  3686400  =  2304000000 

1  =   640  =   6400  =  102400  =   6400000 

1  =    10  =    100  ==    100000 

1.  The  A^cve  is  the  unit  of  land  measure. 

2.  Government  lands  are  divided  into  Townships,  by  parallels  and 
meridians,  each  containing  36  square  miles  or  Sections.  Each  sec- 
tion contains  640  acres  (1  sq.  mile),  and  is  subdivided  into  half-sec- 
tionSy  qua/rier -sections,  etc. 

3.  Measurements  of  land  are  commonly  recorded  in  square  miles, 
acreSy  and  hundredth's  of  an  acre.     The  rood  is  no  longer  used. 

CUBIC    OR    SOLID    MEASURE. 

17.  Cubic  or  Solid  Measure  is  used  in  com- 
puting the  contents  of  solids ;  as  timber,  wood,  stone, 
boxes  of  goods,  the  capacity  of  rooms,  etc. 

Table. 

1728  Cubic  la  (C2^.m.)=l  Cubic  Ft.,  cu.fi.  L '^^  ^_  j  46656  cw.iX 
27  Cubic  Ft.  =1  CnhicYd.,  cu.yd.]  (       %1cu.fi. 

1.  A  Begist&r  Ton,  used  in  measuring  the  entire  internal  capacity 
or  tonnage  of  vessels,  is  100  cubic  feet. 

2.  A  Shipping  Ton,  used  in  measuring  ca/rgoes,  is  40  cubic  feet  in 
the  U.  S.^  and  in  England  42  cubic  feet. 


238  MEASURES     OF     EXTENSION. 

18.   Wood  Measure  is  used  to  measure  wood  and 

rough  stone. 

Table. 

16    Cubic  Feet  =  1  Cord  Foot    ....    cd.ft. 


^\    r=  ICord Cd. 

128    Cubic  Feet       ) 


%    Cord  Feet,  or) 
18    Cubic  Feet       ) 

24^  Cubic  Feet  =  1  \  ^^^^^  °^  ^^^^^^  I   Pch, 

I    or  of  Masonry.    ) 


or  of  Masonry. 

A  Cord  of  wood  is  a  pile  8  ft.  long,  4  ft.  wide,  and  4  ft.  high. 

A  cord-foot  is  1  ft.  in  lengtb  of  such  a  pile ;  that  is,  1  ft.  long, 
4  ft.  wide,  and  4  ft.  Mgh. 

A  Perch  of  stone  or  of  masonry  is  16^  ft.  long,  \\  ft.  wide,  and 
1  ft.  high.     Stone-masons  usually  call  25  cu.  ft.  a  perch. 

19.  Duodecimals  are  the  parts  of  a  unit  resulting 
from  continually  dividing  by  12,  and  are  sometimes  used 
in  measuring  surfaces  and  solids. 

Table. 

12  Fourths  ("")  =  !  Third 
12  Thirds  ~  1  Second 

12  Seconds  =  1  Prime 

12  Primes  =  1  Foot  . 

The  marks  ',  ",  "  ',  "  ",  are  called  indices. 

Railroad  and  transportation  companies  estimate  light  freight  by 
the  space  it  occupies  in  cubic  feet,  but  heavy  freight  by  weight. 

Masonry  is  estimated  by  the  cubic  foot,  and  perch ;  also  by  the 
square  foot  and  square  yard. 

Materials  are  usually  estimated  by  cubic  measure  ;  the  uork  by 
cubic,  or  by  square  measure. 

Engineers,  in  making  estimates  for  excavations  and  eTrtbanTcmentSy 
take  the  dimensions  with  a  line  or  measure  divided  into  feet  and 
decimals  of  a  foot.  The  computations  are  made  in  feet  and  deci- 
mals, and  the  results  are  reduced  to  cubic  yards.  In  civil  engineer- 
ing, the  cubic  yard  is  the  unit  to  which  estimates  for  excavations 
and  embankments  are  finally  reduced. 


V" 

^  20736' 

1" 

V 

lFt.  =  ^ 

1728' 
144' 

Ft. 

12' 

MEASURES     OF     EXTENSION.  239 

A  cubic  yard  of  common  earth  is  called  a  load. 

Brickwork  is  generally  estimated  by  the  1000  bricks,  sometimes  in 
cubic  feet. 

Bricks  are  of  various  dimensions.  The  average  size  of  a  common 
brick  is  8  in.  long,  4  in.  wide,  and  2  in.  thick. 

Philadelphia  or  Baltimore  front  bricks  are  8^:  x  4^  x  2|  inches  ; 
North  Eiver  bricks,  8 x 3J^ x  2 J  inches  ;  Maine  bricks,  7^  x  3J  x2|; 
and  Milwaukee  bricks,  8^  x  4^  x  2|  inches. 

A  cubic  foot  is  estimated  to  contain  27  bricks  laid  dry.  Laid  in 
mortar,  an  allowance  is  made  of  from  -,^j  to  i  for  the  mortar. 

Five  courses  of  bricks  in  the  height  of  a  wall  are  called  a  foot. 

A  brick  wall  which  is  a  brick  and  a  half  thick  is  said  to  be  of  the 
standard  thickness. 

In  estimating  material,  allowance  is  made  for  doors,  windows, 
and  cornices. 

In  estimating  the  work,  masons  measure  each  wall  on  the  outside. 
Ordinarily,  no  allowance  is  made  for  doors,  windows,  and  cornices, 
but  sometimes  an  allowance  of  one-Mlf  is  made,  this  being  a  matter 
of  contract. 

In  scaling  or  measuring  timber  for  shipping  or  freighting,  I  of 
the  solid  contents  of  round  timber  is  deducted  for  waste  in  hewing 
or  sawing.  Thus,  a  log  that  will  make  40  feet  of  hewn  or  sawed 
timber,  actually  contains  50  cubic  feet  by  measurement ;  but  its 
market  value  is  only  equal  to  40  cubic  feet  of  hewn  or  sawed 
timber. 

Sawed  timber,  joists,  plank,  and  scantlings  are  geijerally  bought 
and  sold  by  what  is  called  board  measure.  Hewn  and  round  timber 
by  cvMc  measure. 

In  board  and  lumber  measure,  all  estimates  are  made  on  1  inch 
in  thickness  ;  in  buying  and  selling  lumber,  one-fourth  the  price  is 
added  for  every  \  inch  thickness  over  an  inch. 

In  Board  Measure  all  boards  are  assumed  to  be  1  in.  thick. 

A  board  foot  is  1  ft.  long,  1  ft.  wide,  and  1  inch  thick  ;  hence  12 
board  feet  make  1  cubic  foot. 

Board  feet  are  changed  to  cubic  feet  by  dividing  by  12. 

Cubic  feet  are  changed  to  board  feet  by  multiplying  by  12. 


C  32  gi. 

1  Gal,  =  ■}    Spt, 


24fO  MEASURES     OF     CAPACITY. 

MEASUEES   OF   CAPACITY. 

30.  Capacity  signifies  extent  of  room  or  space. 

31.  Measures  of  capacity  are  divided  into  two  classes ; 
Measures  of  Liquids  and  Measures  of  Dry  Substances. 

33.  The  Units  of  Capacity  are  the  Gallon  for  Liquid, 
and  the  Bushel  for  Dry  Measure. 

LIQUID    MEASURE. 

33.  Liquid  Measure  is  used  in  measuring  liquids  ; 
as  spirituous  liquors,  oil,  molasses,  milk,  water,  etc. 

Table. 
4  Gills  (gi.)  =  1  Pint    .    .    .  pt. 
2  Pints         =  1  Quart .     .     .  qt. 
4  Quarts       =  1  Gallon     .     .  gcd. 
The  Standard  Liquid  Gallon  of  the  United  States  contains  231 
cubic  inches,  and  is  equal  to  about  8^  lb.  Avoir,  of  pure  water. 

The  Imperial  Gallon  of  Great  Britain  contains  277.274  cubic 
inches,  and  is  equal  to  about  1.2  U.  S.  liquid  gallons. 

The  Old  Ale  or  Beer  Measure  is  out  of  use.  The  gallon  contained 
282  cubic  inches. 

34.  In  estimating  the  capacity  of  cisterns,  reservoirs, 

etc.: 

31i  Gallons  make  1  Barrel    .     .     .    Hbl. 

63    Gallons      "       1  Hogshead  .     .    hhd. 

1.  The  barrel  and  hogshead  are  not  jfixed  measures,  but  vary 
-when  used  for  commercial  purposes,  the  former  containing  from  28 
to  45  gallons,  the  latter  from  60  to  125  gallons. 

2.  In  some  of  the  New  England  States  the  barrel  is  estimated  at 
32  gallons  ;  in  some  States  31^  gallons,  and  in  others  from  28  to  32. 

3.  The  tierce,  hogshead,  pipe,  puncheon,  butt  and  tun  are  the 
name  of  casks,  and  do  not  express  any  fixed  or  definite  measures. 
They  are  usually  gauged,  and  have  their  capacities  in  gallons 
marked  on  them. 


MEASURES  OF  CAPACITY. 


241 


35.  Apothecaries^  Fluid  Measure  is  used  by 
physicians  and  apothecaries  in  prescribing  and  com- 
pounding liquid  medicines. 

Table. 

60  Minims,  or  drops  (TTL  or  gtt,  )  =  1  Fluidraclim    ,.    .  /3  . 
8  Fluidraclims  =  1  Fluidounce     .     .  /I . 

16  Fluidounces  =  1  Pint 0. 

8  Pints  =  1  Gallon    ....  Gong, 

Cong.  1  =  0.  8  =/f  128  =fl  1024  =  Itl  61440. 
0.  is  an  abbreviation  of  octans,  tlie  Latin  for  one-eighth  ;  Gong,  for 
congiarium,  the  Latin  for  gallon. 

A  common  teaspoonful,  or  45  drops,  makes  about  one  fluidrachm. 
A  common  teacup  holds  about  4  fluidounces  ;  a  common  tablespoon, 
about  half  a  fluidounce ;  a  pint  of  water  weighs  a  pound. 

![^is  an  abbreviation  for  recipe ,  or  take  ;  a.,  aa.,  for  equal  quanti- 
ties; j.  for  1 ;  ij.  for  2  ;  ss.  for  senfii,  or  half;  gr.  for  grain;  P.  for 
a  little  part ;  P.  aeq.  for  equal  parts  ;  q.  p.,  as  much  as  you  please. 

36.     COMPARISON     OF 


Country. 
England, 
France, 
Prussia, 
Austria, 
Sweden, 


Measure.  U.  S.  gal, 

1  .Gal.  =1.2 

1  Dl.  =     2.64 

1  Quart  =       .30 

1  Maas  =r       .37 


LIQUID     MEASURES. 

Country. 

Measure. 

U.S.  gal. 

Switzerland, 

1  pot 

=       .40 

Turkey, 

Almud 

=     1.38 

Mexico, 

1  Fasco 

=      .63 

Brazil, 

1  Medida 

=      .74 

Cuba, 

1  Arroba 

=    4.01 

South  Spain 

1  Arroba 

=:     4.25 

1  Kanna    =       .69 
Denmark,       1  Kande    =       .51 

DRY    MEASURE, 

37.  I>ry  Measure  is  used  in  measuring  articles 
not  liquid ;  as  grain,  fruit,  salt,  roots,  etc. 

Table. 


2  Pints  (pt)  =  1  Quart .     . 

.  qt 

{  64:  pi. 

8  Quarts        =  1  Peck    .     . 

.  pk. 

IBu.  = 

•  S2qt. 

4  Pecks         =  1  Bushel     . 

.  bu. 

i    ^Pk. 

242  MEASUKES     OF     CAPACITY. 

The  8tanda/rd  Bushel  of  the  United  States  contains  2150.42  cubic 
inches,  and  is  a  cylindrical  measure  18^  inches  in  diameter  and  8 
.  inches  deep. 

The  half-peck,  or  dry  gallon,  contains  268.8  cubic  inches.  Six 
quarts  dry  measure  are  equal,  to  nearly  7  quarts,  liquid  meas- 
ure. 

The  Imperial  Bushel  of  Great  Britain  contains  2218.192  cu.  in. 

The  English  Quarter  contains  8  Imperial  bushels,  or  8^  U.  S, 
bushels. 

Grain  is  shipped  from  New  York  by  the  Quarter  of  480  lb. 
(8  U.  S.  bu.),  or  by  the  Ton  of  33i  U.  S.  bushels. 

The  bushel,  heap  measure,  is  the  Winchester  bushel,  heaped  in 
the  form  of  a  cone,  not  less  than  6  inches  high  and  19^  inches  in 
diameter,  equal  to  the  outside  diameter  of  the  standard  bushel 
measure,  and  equal  to  2747.715  cu.  in. 

Grain,  seeds,  and  small  fruits  are  sold  by  stricken  measure,  or  the 
measure  must  be  even  full. 

Corn  in  the  ear,  potatoes,  coal,  large  fruits,  coarse  vegetables 
and  other  bulky  articles,  are  sold  by  heap  measure. 

It  is  suflBiciently  accurate  in  practice  to  call  5  stricken  measures 
equal  to  4  heaped  measures. 

The  value  of  many  kinds  of  grain,  seeds,  fruit,  and  other  articles, 
are  often  determined  by  weight  instead  of  by  bulk. 

American  coal  is  bought  and  sold,  in  large  quantities,  by  the  ton; 
in  small  quantities,  by  the  bushel. 

The  liquid  and  dry  measures  of  the  same  denomination  are  of 
different  capacities.  The  exact  and  the  relative  size  of  each  may  be 
readily  seen  by  the  following 

38.     COMPARATIVE    TABLE    OF    MEASURES    OF 
CAPACITY. 


Cubic  in.  in 

Cubic  in.  in 

Cubic  in.  in 

Cubic  in.  in. 

one  gallon. 

one  quart. 

one  pint. 

one  gill. 

Liquid  measure .     .   v 

.     .    231 

mi- 

385 

rh 

Dry  measure  (J  pk.) 

.    .    2681 

67J 

331 

81 

A  cubic  foot  of  pure  water  weighs  1000  oz  ,  62  V  lb.  Avoir. 


MEASUEES     OF     CAPACITY. 

243 

39. 

COMPARISON   OF  GRAIN   MEASURES 

Country. 

Measure.        U.  S.  bush. 

Country. 

Measure. 

U.  S.  bush. 

England, 

1  Bushel        =   1.031 

Germany, 

1  Schef. 

=  1.5  to  3 

France, 

1  Hectoliter  =  2.84 

Persia, 

1  Artaba 

=  1.85 

Prussia, 

1  Scheffel      =  1.56 

Turkey, 

IKilo 

=  1.03 

Austria, 

1  Metze         .=  1.75 

Brazil, 

IFan. 

=  1.5 

Russia, 

1  Chetverik  =     .74 

Mexico, 

1  Alque. 

=  1.13 

Greece, 

1  Kailon        =  2.837 

Madras, 

1  Parah 

=  1.743 

ENGLISH  MEASURES  OF  CAPACITY. 

30.   Wine  Measure  is  used  to  measure  wines  and 
all  liquids,  except  malt  liquors  and  water. 


Table. 

4  Gills 

=   1  Pint     .... 

pU 

2  Pints 

=  1  Quart .... 

gt. 

4  Quarts 

=  1  Gallon      .     ,     . 

gal. 

10  Gallons 

=  1  Anker      .     .    . 

ank. 

18  Gallons 

=  1  Runlet     .     .    . 

run. 

43  Gallons 

=  1  Tierce  .... 

tier. 

2  Tierces 

=  1  Puncheon     .     . 

pun. 

63  Gallons 

=  1  Hogshead     .    . 

Mid. 

2  Hogsheads 

=  1  Pipe    .... 

pipe. 

2  Pipes 

=  1  Tun    ...     . 

tun. 

31.  Ale  and  Seer  Measure  is 

used  to  measure 

all  malt  liquors  and  water. 

Table. 

2  Pints 

=  1  Quart  .     .     . 

.    qt. 

4  Quarts 

=  1  Gallon      .     . 

.    gal. 

9  Gallons 

=  1  Firkin.     .     . 

.    fir. 

18  Gallons 

=  1  Kilderkin 

.    kU. 

86  Gallons 

=  1  Barrel .     .     . 

bar. 

li  Bar.  or  54  gal 

.  =  1  Hogshead     . 

hhd. 

2  Hogsheads 

=  1  Butt     .     .     . 

butt. 

2  Butts 

=  1  Tun 

tun. 

244 


MEASURES     OF     WEIGHT. 


33.  Corn  or 

Dry  Measure 

is  used  to  measure 

all  dry  commodities  not  usually  heaped 

r^ 

Table. 

2  Quarts 

=  1  Pottle  .    . 

.    pot. 

2  Pottles 

=  1  Gallon  .    .     . 

.    gal. 

2  Gallons 

=  1  Peck     .    . 

.    ph. 

4  Pecks 

=  1  Bushel      . 

.    lus. 

2  Bushels 

=  1  Strike   .     . 

.    str. 

4  Bushels 

r=  1  Coomb  .     . 

.    .    coornb. 

2  Coombs  or  8  bu.=  1  Quarter     . 

.     .     qr. 

5  Quarters 

=  1  Load     .     . 

.    .    load. 

2  Loads  or 

10Qr.=  lLast      .     . 

.    last. 

14    Pounds  =  1  Stone. 
21i  Stones    —  1  Pig  of  iron  or  lead. 
8    Pigs        =:  1  Pother. 
The  stone  varies.     Legally  it  is  14  lb.     A  stone  of  butcher's  meat 
or  fish  is  reckoned  at  8  lb.  ;  of  cheese,  at  16  lb. ;  of  hemp,  at  32  lb. 
Kpig  of  iron  or  lead  is  250  lb.,  and  8  pigs  make  2i  father. 


MEASURES   OF  WEIGHT. 

33.  Weight  is  the  measure  of  the  quantity  of  matter 
a  body  contains,  determined  by  the  force  of  gravity. 

34.  The  Standard   Unit  of  weight  is  the  Tro^ 
Pound  of  the  Mint,  and  contains  5760  grains. 

TROY    ^VEIGHT, 

35.  Troy  Weight  is  used  in  weighing  gold,  silver, 
jewels,  and  in  philosophical  experiments. 

Table. 

24  Grains  {gr)        =  1  Pennyweight     .  pwt.  (  5760  gr. 

20  Pennyweights    =  1  Ounce  .     .     .     .  oz.         1  ^.  =  -<    240  pwt 

12  Ounces  =  1  Pound  ....  ^6.     1  {      12  oz. 


measures    of    weight.  245 

36.  Table. 


DIAMOND  WEIGHT. 


16  Parts  —  1  Grain. 
4  Grains  =  1  Carat. 
1  Carat     =  3i  Troy  gr.,  nearly. 


ASSAYERS*  WEIGHT. 


1  Carat        =10  pwts. 
1  Carat  gr.  =  2  pwts.  12  gr.  or 
60  Troy  gr. 
24  Carats      =  1  Troy  lb. 
The  term  carat  is  also  used  to  express  the  fineness  of  gold,  each 
carat  meaning  a  twenty-fourth  part. 

APOTHECARIES*  ^VEIGHT. 

37.  Apothecaries^  Weight  is  used  by  apotheca- 
ries and  physicians  in  compounding  dry  medicines. 

Table. 

20  Grains  {gr,  xx)  =  1  Scrapie  .     .    .     .  sc,    or  3. 

3  Scruples  (3 iij)  =  1  Dram dr.,  or  3. 

8  Drams  ( 3  viij)    =  1  Ounce      .     .     .     .  <>g.,    or  g . 

12  Ounces  (.§  xij)    =  1  Pound     .     .     .    .  lb.,    or  a. 

rt,l  =  ll2  =  zm=^2SS=gr.  5760. 

1.  Medicines  are  bought  and  sold  in  quantities  by  Avoirdupois 
weight. 

2.  The  pound,  ounce  and  grain  are  the  same  as  those  of  Troy 
weight,  the  ounce  being  differently  divided. 

AVOIRDUPOIS  W^EIGHT. 

38.  Avoirdupois  Weight  is  used  for  all  the  ordi- 
nary purposes  of  weighing. 


Table. 

16  Ounces  (oz.)         =  1  Pound      .    .    .  lb. 

100  Pounds    ,  =1  Hundredweight  cwt. 

20  cwt.,  or  2000  lb.  =  1  Ton    .     .     .     .  T. 


i  32000  oz. 
2000  lb. 
20  ewe. 


The  oun£e  is  often  divided  into  halves,  quarters,  etc. 


246 


MEASURES     OF     WEIGHT. 


39.  The  Long  or  Gross  ton,  hundred-weight,  and 
quarter  were  formerly  in  common  use  ;  but  they  are  now 
seldom  used  except  in  estimating  duties  at  the  United 
States  Custom-Houses,  in  freighting  and  wholesaling 
coal  from  the  Pennsylvania  mines. 


38  Pounds 

4Qr.,  or  112  lb. 
20  cwt.  or  3240  lb. 


LONO    TON    TABLE. 

—  1  Quarter  .  .  .  qr. 
=  1  Hundredweight  cwt. 
=  1  Ton T. 


1  T. 


2240  lb. 
SOqr. 
2t)ci^. 


40.  The  following  denominations  are  also  used  : 
100  Pounds  of  Grain  or  Flour  make  1  Cental. 


100  Pounds  of  Dry  Fish 
100  Pounds  of  Nails 
196  Pounds  of  Flour 
200  Pounds  of  Pork  or  Beef 
280  Pounds 


41. 


TABLE    OF 


1  QuintaL 
IKeg. 
1  Barrel. 
1  Barrel. 

1  Barrel  of  Salt  at  the 
N.  Y.  Salt  Works. 
WEIGHTS. 


COMPARATIVE 

Troy.  Apothixjaries.  Avoirdupois. 

1  Pound  =  5760  Grains     =  5760  Grains     =7000    Grains. 
1  Ounce  =    480      '*         =    480      ''         =    437.5      " 

175  Pounds  =    175  Pounds   =    144    Pounds. 


42.  COMPARISON 

Weight.    U.  S. 
1  KHogram  r 


OF   COMMERCIAL  ^A^EIGHTS. 


Country, 
France, 
Germany, 
Austria, 
Russia, 
Sweden, 
Denmark, 
Turkey, 
Egypt, 
Persia, 
Madras, 


1  Pfund 
1  Pfund 
1  Funt 
IPund 
IPund 
1  Oka 
1  Rottoli, 
1  Battel, 
IVis 


lbs.  avdp. 

=  2.20 

=  1.10 

=  1.23 

=  .90 

=  .93 

=  1.10 

=  2.82 

=  1.008 

=  2.116 

=  3.125 


Weight. 


Country. 

Prussia,         1  Zolpf'd 
Netherlands,!  Pond 
East  Indies,  1  Seer 


U.  S.  lbs.  avdp. 
l.IO 


China, 

Japan, 

Mexico, 

Brazil, 

Spain, 

Sicily, 

Arabia, 


1  Catty 
IKin 
1  Libra 
1  Arratel 
1  Libra 
1  Libra 
1  Maund 


=  2.20 

=  2.06 

=  1.33 

=  .63 

=  1.02 

=  1.02 

=1.016 

=  .7 

=  .8 


MEASURES     OF     WEIGHT. 


247 


43.  The  weight  of  the  bushel  of  certain  grains,  seeds 
and  vegetables  has  been  fixed  by  statute  in  many  of  the 
States  ;  and  these  statute  weights  must  govern  in  buying 
and  selling,  unless  specific  agreements  to  the  contrary  be 
made. 

TABLE  OF  AVOIRDUPOIS  POUNDS  IN  A  BUSHEL, 

As  prescribed  hy  statute  in  the  several  States  named. 


COMMODITIES. 


Barley 

Beans 

Blue  Grass  Seed... 

Buckwheat 

Castor  Beans 

Clover  Seed  

Dried  Apples 

Dried  Peaches.  ... 

Flax  Seed 

Hemp  Seed 

Indian  Corn . . 

Indian  Com  in  ear. 
Indian  Corn  Meal. . 

Oats 

Onions    

Potatoes 

Rye 

Rye  Meal 

Salt 

Timothy  Seed 

Wheat 

Wheat  Bran 


50 


40 


45 


32 


54 


60 


56 


56 


t^ 


^^ 


48  48 
60  60 
14!  14 
52  52 
46 


32 


14 
50 
46 
60 
2524 


60 


56 

44 

56 

68 

50 

|32 

57  48 

60  60 

54  56 


56 
44 
56 

50 

33>^ 

57 
60  60 
56156 


50  50  50 
45  45'45;45 
60  60  60|60 

20|     |20,2a 


56 


60 


)48 


46 


42 


48 


42 


28 


56 


.1 


50 

30  32  32 

52 
60 

50 


56  56 

50 


60 


60 


48 
60 
14 
52 

4() 
CO 
24 
3b 

;56 

J44 
56  52 


56 


50 

45 

60,60 


48'48 


60  f 


55  55  56 

'  I 

56  58  56 
30  32  32 

I  I 

60  60: 
56  56  56 

u 

'44! 
0606 

I     I 


47 


56 


32 


56 


46 


46 


50 


50 


56 


60 


45 


42 


28 


56 


46 

60  60 


In  Pennsylvania  80  lbs.  coarse,  70  lbs.  ground,  or  62  lbs.  fine  salt 
make  1  bushel ;  and  in  Illinois,  50  lbs.  common  or  55  lbs.  fine  salt 
make  1  bushel. 

In  Maine  64  lbs.  of  ruta  baga  turnips  or  beets  make  1  bushel. 

A  cask  of  lime  is  240  lbs.  Lime  in  slaking  absorbs  2i  times  its 
volume,  and  2i  times  its  weight  in  water. 


248 


MEASURES     OF     WEIGHT. 


44,  The  following  table  will  assist  farmers  in  making 
an  accurate  estimate  of  the  amount  of  land  in  different 
fields  under  cultivation. 

Table. 

A. 


10  rods   X     16 

rods 

=    1   A. 

220 

feet 

X 

198    feet   =  1 

8    "       X     20 

« 

=    1    " 

110 

" 

X 

369       "      =  1 

5    "       X     32 

** 

r=     1     " 

60 

a 

X 

726      "      =  1 

4    "       X     40 

" 

,  1      ti 

120 

t( 

X 

363       "      =  1 

5  yds.     X   96S 

yds. 

1     '' 

200 

<( 

X 

108.9    "      =  i 

10    "       X   484 

it 

1      " 

100 

*< 

X 

U5.2    ''      =  4 

20    "       X   242 

<( 

=     1      " 

100 

" 

X 

108.9    "      =  i 

40    ''       X    121 

" 

=     1     "    ' 

45.  The  following  table  will  often  be  found  convenient, 
taking  inside  dimensions : 

A  box  24  in.  x  24  in.  x  147  will  contain  a  barrel  of  31  i  gallons. 

A  box  15  in.  x  14  in.  x  11  in.  will  contain  10  gallons. 

A  box  8}  in.  x  7  in.  x  4  in.  will  contain  a  gallon, 

A  box  4  in.  x  4  in.  x  3.6  in.  will  contain  a  quart. 

A  box  24  in.   x  28  in.  x  16  in.  will  contain  5  bushels. 

A  box  16  in.   x  12  in.  x  11.2  in.  will  contain  a  bushel, 

A  box  12  in.  x  11.2  in.  x  8  in,  will  contain  a  haZf-bu^hel, 

A  box  7  in.  x  0.4  in.  x  12  in.  will  contain  a  peck. 

A  box  8. 4  in.  x  8  in.  x  4  in.  will  contain  a  half -peck  or  4  dry  quarts. 

A  box  6  ill.  by  5 J  in.,  and  4  in.  deep,  will  contain  a  half-gallon. 

A  box  4  in.  by  4  in.  and  2,\  in.  deep,  will  contain  a  pint. 

46.  Nails  are  put  up  100  pounds  to  the  keg. 


.a  r£ 

«jQ 

■2  aJ 

Oi  ,£5 

^  00 

a&  jD 

Size. 

fl 

3   °3 

Size. 

il 

?'S 

Size. 

§.s 

3.9 

^.S 

5.9 

^.9 
T6 

3.9 

^.3 

3cZfineblaed. 

U 

725 

30c?  com.  blued. 

4i 

M  casing 

2 

210 

3c?  com.    " 

u 

403 

40c?    " 

5 

14 

8c?       '' 

2i 

134 

Ad    "       " 

li 

300 

50c?    " 

5i- 

11 

10c?       *'       • 

3 

78 

M    ''       " 

2 

150 

60c?    " 

6 

8 

6c?  finishing 

2 

317 

M    "       '* 

2.^ 

85 

6c?  fence. 

2 

80 

8c?      " 

2^ 

208 

10^  "       *' 

3 

60 

8c?    " 

2J- 

50 

10c?      '* 

3 

126 

12d  "       " 

Sj 

50 

10c?    " 

3 

30 

6c?  clinching 

2 

118 

16c?  "       " 

3i 

40 

12c?    " 

31 

27 

8c?      '' 

2i 

80 

20cZ  '' 

4 

20 

16c?    « 

3i 

20 

10c?      '' 

3 

45 

5  lbs.  of  4c?  or  3J  lbs.  of  3c?  will  put  on  1,000  shingles. 
5f  lbs.  of  3c?  fine  will  put  on  1,000  lath. 


MEASURES    OF    TIME    AND    ANGLES. 


249 


MEASURES    OF    TIME. 

47.  Time  is  the  measure  of  duration. 

48.  The  Unit  is  the  mean  solar  day  . 

Table. 


CO  Seconds  {sec.) 
60  Minutes 
24  Hours 
7  Days 
365  Days,  or          ) 
12  Calendar  Mo. ) 

=  1  Minute    .    .    min. 
=  1  Hour  ,    .     .    hr. 
=  1  Day    .     ,     .    da. 
=  1  Week      .     .    wk. 

=  1  Common  Year  yr. 

Common  Year. 
r  525600  min. 
I  Yr.=   \       8760  ;^r. 
1           12  Tno. 

366  Days 
100  Years 

=  1  Leap  Year   .    yr. 
=  1  Century  .     .     Cen. 

VW     lA/lA/. 

1.  Every  year  that  is  exactly  divisible  by  4  is  a  leap  year,  the 
centennial  years  excepted  ;  the  other  years  are  common  years. 

2.  Every  centennial  year  that  is  divisible  by  400  is  a  leap  year. 

3.  In  most  business  transactions  30  days  are  called  1  month,  and 
12  months  1  year. 

4.  The  civU  day  begins  and  ends  at  12  o'clock,  midnight.    A.  M. 
denotes  the  time  before  noon ;  M.,  at  noon  ;  and  P.  M.,  afternoon. 

5.  The  astronomical  day,  used  by  astronomers  in  dating  events, 
begins  and  ends  at  12  o'clock,  noon. 


MEASURE    OF    ANGLES. 


49.  Circular  or  An- 
gular Measure  is  used 
in  measuring  angles  and  arcs 
of  circles,  in  determining  lat- 
itude and  longitude,  the  loca- 
tion of  places,  the  motion  of 
the  heavenly  bodies,  etc. 


/ 

^  1296000". 

o 

1(7.  =    ^ 

21600' 
360^ 

a 

12  8. 

250        MEASUEES    OF    TIME    AND    ANGLES. 

50.  The  Unit  is  the  degree,  which  is  -^  part  of  the 
circumference  of  any  circle. 

Table, 

60  Seconds  (")      =1  Minute  . 
60  Minutes  =  1  Degree  . 

30  Degrees  =  1  Sign  .    . 

12  Signs,  or  360°  =  1  Circle    . 

A  Seini-Circumf^ce  is  one-half  of  a  circumference,  or  180°. 

A  Quadrant  is  one-fourth  of  a  circumference,  or  90°. 

A  Sextant  is  one-sixth  of  a  circumference,  or  60°. 

A  Sign  is  one-twelfth  of  a  circumference,  or  30''. 

A  Degree  (1°)  is  one-ninetieth  of  a  right  angle. 

The  length  of  a  degree  varies  with  the  size  of  the  circle  ;  thus, 
a  degree  of  longitude  at  the  Equator  is  69.16  statute  miles,  at  30° 
of  latitude  it  is  59.81  miles,  at  60°  of  latitude  it  is  34.53  miles,  and 
at  90°,  or  the  poles,  it  is  nothing. 

A  mm-z^^g  of, the  earth's  circumference  is  called  a  geographic  mile. 

LONGITUDE    AND    TIME. 

51.  Since  the  earth  performs  one  complete  revolution 
on  its  axis  in  a  day  or  24  hours,  the  sun  appears  to  pass 
from  east  to  west  round  the  earth,  or  through  360°  of 
iongitude,  once  in  every  24  hours  of  time.  Hence  the  re- 
lation of  time  to  the  real  motion  of  the  earth  or  the 
apparent  motion  of  the  sun,  is  as  follows  : 


Table. 

For  a  difference  of 

There  is  a  difference  of 

15°  in 

Long. 

1  hr.   in  Time. 

15'   " 

** 

iTTiin."       " 

15"  " 

(1 

1  sec.  "      « 

1°     " 

(1 

4min."^    " 

1'     " 

n 

4  sec.  "      '* 

1'    •* 

it 

A  sec.  -      •* 

MISCELLANEOUS. 


251 


COUNTING. 

53.  This  measure  is  used  in  counting  certain  classes 
of  articles  for  market  purposes. 

Table. 


12  Units 

=  1  Dozen    .     . 

doz. 

C  1728  vnits. 

12  Dozen 

=  1  Gross     .     . 

.    gro. 

1  G.  gro.  = 

I    lUdoz. 

12  Gross 

=  1  Great  Gross 

0.  gro. 

i      12  gro. 

20  Units 

=  1  Score      .     . 

.    sc. 

Two  things  of  a  kind  are  often  called  a  pair,  and  six  things  a 
set;  as  Sipair  of  horses,  a  set  of  chairs,  etc, 

PAPER. 

53.  The  denominations  of  this  table  are  used  in  the 
paper  trade 

Table. 


24  Sheets 

=  1  Quire    . 

.     qr. 

'  4800  Sheets. 

20  Quires 
2  Reams 

=  1  Ream    . 
=  1  Bundle . 

.    rm. 
.     hurt. 

1B.=  ^ 

200  Quires. 
10  Reams. 

5  Bundles 

-  1  Bale      . 

.    B. 

5  Bundles. 

Paper  is  bought  at  wholesale  by  the  bale,  bundle,  and  ream ;  and 
at  retail  by  the  ream,  quire,  and  sheet. 

Paper  may  be  made  to  order  of  any  size,  but  the 
greater  part  made  up  for  sale  is  only  of  regular  sizes. 
The  names  generally  define  the  sizes.  Writing  and  Draw- 
ing Papers  differ  in  size  from  Printing  Papers  of  the 
same  name,  English  sizes  differ  from  American.  How- 
eyer,  when  English  or  French  printing  papers  are  made 
for  this  country  they  are  of  American  sizes. 


252 


MISCELLAKEOUS. 


54.  SIZE    OF   WRITING    PAPERS, 

FOLDED    PAPERS. 


Inches. 

Xnches, 

Billet  Note  .... 

.6x8 

Letter 

10    xl6 

Octavo  Note     .     .     . 

.7x9 

Commercial  Letter 

11    xl7 

Commercial  Note     . 

.    8   xlO 

Packet  Post.     .     . 

lUxl8 

Packet  Note    .     .     . 

.    9   xll 

Extra  Packet  Post 

llixlSi 

Bath  Note  .... 

.    8ixl4 

Foolscap .... 

m  X 16 

The  dimensions  given  above  are  those  in  most  general 
use.     Some  kinds  occasionally  vary  a  trifle. 


55. 


FLAT    PAPERS. 


Law  Blank  .... 

Inches. 
13x16 

Flat  Cap      .... 

14x17 

Crown 

15x19 

Demy 

16x21 

Folio  Post    .... 

17x22 

Check  Folio     .     .     . 

17x24 

Double  Cap      .     .     . 

17x28 

Extra  Size  Folio  .     . 

19x28 

Inches. 

Medium 18    x  23 

Royal 19    x24 

Super  Royal  .  . 
Imperial .... 
Elephant  .  .  . 
Columbia      .     .     . 

Atlas 

Double  Elephant  . 


20   x28 

22  x30 

22\ X  27} 

23  x33J- 
26  X  33 
26   X   40 


Extra  Size  Folio  is  sometimes   18  x  23   inches,  and 
19  X  24  inches.     Imperial  is  sometimes  93  x  31  inches. 


SIZE    OF     PRINTINQ     PAPERS. 


56. 

Inches. 

Medium 19  x  24 

Royal 20  X  25 

Super  Royal    ....  22x28 

Imperial 22  x  32 

Medium-and-half      .     .  24x30 

Small  Double  Medium .  24  x  36 


Double  Medium  .     . 

Double  Royal  .     .     . 

Double  Super  Royal 

<(  <(  (( 

Broad  Twelves     .     . 
Double  Imperial  .     . 


Inches. 
24x38 
26x40 
28x43 
29x43 
23x41 
32x46 


Larger  sizes  and  odd  sizes  are  sometimes  made,  but  are 
not  common. 


MISCELLANEOUS. 


253 


BOOKS. 

57.  The  terms  folio,  quarto,  octavo,  duodecimo,  etc., 
indicate  the  number  of  leaves  into  which  a  sheet  of  paper 
is  folded. 


When  a  sheet  is 
foldea  into 

2  leaves 

4      " 

8      '* 

The  book  is 
called 

a  Folio, 

a  Quarto  or  4to, 

an  Octavo  or  8vo.  • 

And  1  sheet  of 
paper  makes 

4  pp.  (pages). 

8   '' 
16   '' 

12      " 

a  Duodecimo  or  12mo, 

24   - 

16      " 

a  16m,o, 

82   *' 

18     " 

24     "          . 

32      *' 

an  18mo, 
a  24mo, 
a  32mo, 

36   '' 
48   " 
64  '^ 

COPYING. 
58.  Clerks  and  copyists  are  often  paid  by  i\\Q  folio  for 
making  copies  of  legal  papers,  records,  and  documents. 

72  words  make  1  folio,  or  sheet  of  common  law. 
90  "  1  '*  chancery. 

A  folio  varies  in  different  States  and  countries,  but  usually  con- 
tains from  75  to  100  words. 


59. 


ROMAN    LONG    MEASURES. 


Digit  .  .  . 
Uncia  (inch) 
Pes  (foot; 


Inches. 

^       .72575 
=      .967 

=r    11.604 


Feet.  Inches. 

Cubit 1       5.408 

Passus    .....    4      10.02 
Mile  (millarium)     4842 


6^0.        JEWISH    LONG    MEASURES. 

Feet. 

Cubit =  1.824 

Sabbath  day's  journey  =    3648 


MUe  (4000  cubits)  . 
Day's  journey  - 


Feet. 
.  =  7296 
33.164  mi. 


61. 


MISCELLANEOUS. 

Feet. 


Arabian  foot  .  .  .  =  1.095 
Babylonian  foot  .  .  =  1.140 
FiO-vTitian  finp-er      .     .   =      .06145 


Feet. 

Hebrew  foot  .  .  .  =  1.213 
cubit  ...  =  1.817 
sacred  cubit .     =  2.002 


254 


MISCELLANEOUS. 


RAILROAD    FREIGHT. 
Q2m  When  convenient  to  weigh  them,  all  goods  are 
billed  at  actual  loeight;  but  ordinarily,  the  articles  named 
below  are  billed,  at  the  rates  giyen  in  the  following     ^ 

Table. 


Ale  or  Beer, 

820  lbs 

per  bbl. 

Highwines, 

350  lbs.  per  bbL 

Apples,  green. 

150 

ft 

« 

Lime, 

200 

f(                   u 

Beef, 

820 

tt 

ff 

:   Nails, 

108 

"  per  keg. 

Barley, 

48 

it 

per  bu. 

Oil, 

400 

"    per  bbL 

Beans, 

60 

(f 

tt 

Oats, 

32 

**    per  bu. 

Cider, 

350 

it 

per  bbl. 

Pork, 

320 

"    per  bbL 

Com  Meal, 

220 

ti 

(f 

Potatoes,  com'n,  150 

((          ti 

Corn,  shelled 

56 

t( 

per  bu. 

Salt,  fine, 

300 

tt         tf 

Corn  in  ear. 

70 

ft 

tt 

•*     coarse. 

350 

ft         tt 

Clover  Seed, 

60 

ft 

" 

"    in  sacks. 

200 

*'  per  sack. 

Eggs, 

200 

ft 

per  bbl. 

Wheat, 

60 

'*  per  bu. 

Fish, 

800 

tc 

t( 

Whiskey, 

350 

"  per  bbL 

Flour, 

200 

ft 

u 

2000  pounds  are  reckoned  1  tan^ 

Generally  from  18000  to  20000  pounds  is  considered  a  car  load. 

63.  Lumber  and  some  other  articles  are  estimated  aa 
follows : 

Amount  for 
Weight.  car  load. 

Pine,  Hemlock,  and  Poflab,  thoroughly 

seasoned,  per  thousand  feet     .     .     .     *     .  SOOO  6500 
Black  Walnut,  Ash,  Maple,  and  Cherry, 

per  thousand  feet 4000  5000 

Pine,  Hemlock,  and  Poplab,  green,  per  M.  4000  6000 
Black  Walnut,  Ash,  Maple,  and  Cherry, 

green,  per  M 4500  4000 

Oak,  Hickory,  and  Elm,  dry,  per  M.  .    .     .  4000  5000 

Oak,  Hickory,  and  Elm,  green,  per  M.    .    .  5000  4000 

Shingles,  green,  per  thousand 375  55  M. 

Lath,  per  thousand 500  40  M 

Brick,  common,  per  car  load 4  lbs.  each.  5000 

Coal,  per  car  load 250  bu. 


MISCELLANEOUS.  255 

64.       SCRIPTURE     LONG     MEASURES. 

Eng.mi.        Paces.  Feet,  Inches. 


A  Palm 

equals 

0 

0 

0 

3.648 

A  Span 

« 

0 

0 

0 

10.944 

A  Fathom 

0 

0 

7 

3.552 

EzekieFs  reed 

0 

0 

10 

11.328 

An  Arabian  pole 

0 

0 

14 

7.104 

A  Furlong 

0 

145 

4.6 

.00 

An  Eastern  mile 

1 

403 

1.0 

.00 

A  Day's  Journey 

33 

172 

4.0 

.00 

65.   SCRIPTURE  MEASURES  OF  CAPACITY. 

LIQUID.  DKY. 


gal. 

pints. 

ACaph 

=      0 

.625 

A  Gachal 

A  Log 

=      0 

.83% 

AKab 

AKab 

=      0 

8.333 

An  Omer 

AHin 

=    1 

2 

A  Seah 

A  Seah 

=:         2 

4 

An  Epah 

A  Bath  or 

Ephah 

=      7 

4 

ALetek 

A  Homer 

=    75 

5 

A  Homer 

ecks 

.      pints. 

0 

.1416 

0 

2.8333 

0 

5.1 

1 

1 

3 

3 

6 

0 

32 


66.    MONEY  MENTIONED  IN   SCRIPTURE. 

£      s.  d.  $    ct8. 

A  Talent  (gold) equal  5464    5    8    ==26592.809 

A  Talent  (silver)     ......  «  341  10    4     =  1662.0249 

A  Manch  or  Mina "  5  13  10     =       27.6990 

A  Pound  (Mina) «  3    4    7     _       15.7151 

A  Shekel  (gold) ^*  1  16    5    =        8.8612 

A  Shekel  (silver) "  q    2    3     —        0.5474 

A  Golden  Daric  or  Dram     .    ,     .  "  1     1  10     =:        5.3127 

A  Piece  of  Silver  (Stater)    .         .  "  0    2    7=        0.6285 

Tribute  Money  (Didrachm)      .     .  "  0    1     3J  =:        0.3142 

^^B^^ah "  0    1    1     =r        0.2636 

A  Piece  of  Silver  (Drachm)     .     .  "  0    0    7J  =        0.1571 

A  Penny  (Denarius) "  0    0    7i  =         0.1520 

A^erah «  0    0    1=        0.0202 

A  Farthing  (Assarium)    ....  "  0    0    0|  =        0.0076 

A  Mite «  0    0    0A=        0.0019 


256 


MONEY. 


MEASURES    OF    VALUE. 

67.  Money  is  the  measure  of  the  value  of  things  or 
of  services,  and  the  medium  of  exchange  in  trade. 


UNITED  STATES  MONEY. 

68.  United  States  Money  is  the  legal  currency 
of  the  United  States. 


69.  The  Unit  of  United 
States  Money  is  the  Gold 
Dollar. 

Table. 

10  Mills  (m.)  —  1  Cent  .  .  ct. 

10  Cents  =  1  Dime  .  .  d, 

10  Dimes  =  1  Dollar  .  $. 

10  DoUajs  =  1  Eagle  .  E. 


\E,  = 


10000  m. 
1000  ct. 
100  d. 
10   $. 


70.  The  legal  Coin  of  the  United  States  consists  of 
gold,  silver,  nickel,  and  bronze,  and  is  as  follows: 

71.  Gold.    The  double-eagle,  eagle,  half-eagle,  quar- 
ter eagle,  three-dollar,  and  one-dollar  pieces. 

73.  Silver.    The   dollar,  half-dollar,  quarter-dollar, 
the  twenty-cent,  and  the  ten-cent  pieces. 

73.  Nickel.    The  five-cent,  and  three-cent  pieces. 

74.  Bronze.    The  one-cent  piece. 

1.  The  half -dime  and  three-cent  pieces,  the  bronze  two-cent,  and 
Hie  nickel  one-cent  pieces  are  no  longer  coined. 


MONEY.  257 

2.  The  Trade-dollar  weighs  420  grains,  and  is  designed  soleiy  for 
purposes  of  commerce  and  not  for  currency.  The  legal-tender  dollar 
weighs  412|  grains. 

3.  The  Standard  Purity  of  the  gold  and  silver  coins  is  .9  pure 
metal,  and  .1  alloy.  The  alloy  of  gold  coins  is  silver  and  copper ; 
the  silver,  by  law,  not  to  exceed  -^^  of  the  whole  alloy.  The  alloy 
of  siUer  coins  is  pure  copper. 

4.  The  five-cent  and  three-cent  pieces  are  composed  of  f  copper 
and  \  nickel.  The  cent  is  composed  of  95  parts  of  copper  and  5 
parts  of  tin  and  zinc. 

CANADA    MONEY. 

75.  Canada  Money  is  the  legal  currency  of  the 
Dominion  of  Canada.  The  denominations  are  dollars^ 
cents,  and  mills,  and  have  tlie  same  nomi7ial  value  as  the 
corresponding  denominations  of  U.  S.  Money. 

The  Currency  of  the  Dominion  of  Canada  was  made  uniform  July 
1st,  1871.    Previous  to  1858  sterling  money  was  in  use. 

76«  The  Coin  of  the  Dominion  of  Canada  is  silver 
and  bronze. 

77.  The  Silver  Coins  are  the  fifty-cent,  twenty- 
five-cent,  ten-cent,  and  five-cent  pieces. 

78.  The  JBronze  Coin  is  the  one-cent  piece. 

The  standard  silver  coins  consist  of  925  parts  (.925) 
pure  silver  and  75  parts  (.075)  copper.  That  is,  they  are 
.925  fine. 

1.  The  gold  coin  used  in  Canada  is  the  British  Sovereign,  worth 
$4.86|,  and  .the  Half- Sovereign. 

2.  The  intrinsic  value  of  the  50-cent  piece  in  United  States 
money  is  about  46 1  cents,  of  the  25-cent  piece  23^  cents.  In  ordi- 
tiary  business  transactions,  they  pass  the  same  as  U.  States  coin. 


258, 


MONEY. 


ENGLISH    MONEY. 

79.  English   or  Sterling  Money   is  the  legal 
currency  of  Great  Britain. 


80.  The  Vnit  of 

English  Money  is  the 
Sovereign,  or  Pound 
Sterling. 


The  value  of  a  Sovereign  in  United  States  Money  is  $4.8665. 
Table. 


4  Farthings  (far)  =     1  Penny  .    .   .   .   d 
13  Pence  =     1  ShiUing    .    .    .   s. 

1  Sovereign,  or  .   sov. 
Pound   ....£. 


20  Shillings 


w 


U.  S.  Value. 

r  .02+. 

£1  =  ^       .293  +  . 
[  $48665. 


Other  Denominations. 


2  Shillings  («.)     =     1  Florin     .    ,    .    fl. 
5  Shillings  =     1  Crown     .    .    .    cr. 


U.  8.  Value. 

$.48665. 

$1.2166  +  . 


81.  The  Coin  of  Great  Britain  in  general  use  con- 
fiists  of  gold^  silvery  and  copper ,  as  follows  : 

82.  Gold*    The  sovereign,  and  half-sovereign. 

83.  Silver.     The  crown,  half-crown,  florin,  shilling, 
six-penny,  and  three-penny  piece. 

84.  Copper.    The  penny,  half-penny,  and  farthing. 
The  standard  gold  coin  contains  11  parts  pure  gold 

and  1  part  alloy ;  silver  coin  37  parts  pure  silver  and  3 
parts  alloy. 


MOKEY. 

FRENCH    MONEY, 


259 


85,  French  Money  is  the  legal  currency  of  Prance, 
and  is  decimal 


The  Franc  of  the 
Eepublic. 

86.    The    Unit 

of  French  Money  is 
the  Silver  Franc. 

The  Franc  of  the 
Empire. 


The  value  of  a  Franc  in  United  States  Money  is  $.193. 
Table. 

10  Millimes  (m.)  =  1  Centime  .  ,  ,  ct. 
10  Centimes  =  1  Decime  ,  ,  .  dc, 
10  Decimes  =  1  Franc  .    ,   ,   .  fr. 

20  Francs  =  1  Napoleon     .    .   Wap. 

87.  The  Coin  of  France  consists  of  gold,  silver,  and 
bronze,  as  follows  : 

88.  Gold.    The  100,  40,  20,  10,  and  5  franc  pieces. 

89.  Silver.  The  5,  2,  and  1  franc,  the  50  and  the 
25  centime  pieces. 

90.  Bronze.     The  10,  5,  2  and  1  centime  pieces. 
The  standard  gold  and  silver  coins  contain  9  parts  of 

pure  metal  and  1  part  of  alloy. 

The  U.  S.  Congress,  by  the  Act  of  1873,  fixed  the  weight  of  the 
silver  half-dollar  at  12 1  metrical  grammes,  so  that  2  half-dollars  are 
precisely  equivalent  in  value  to  the  5  franc  silver  coin  of  Europe. 


1  N'ap.  =  ^ 


r  20000  m. 
2000  ct. 


200  dc. 
20fr, 


260 


MONEY. 


GERMAN    MONEY. 

91.     The    New   Empire    of  Germany    has 

adopted  a  new  and  uniform  system  of  coinage. 


93.    The   Unit 

of  this  new  German 
System  of  Coinage 
is  the  Reichsmark. 


The  value  of  a  Reichsmark  ('*  Mark'')  in  U.  S.  Money  is  $.2385 

A  pound  of  gold  .900  fine  is  divided  into  139|  pieces,  and  the  ^ 
part  of  this  gold  coin  is  called  a  "Mark,"  and  this  is  subdivided 
into  100  pennies  (Pfennige). 

The  Coin  of  the  New  Empire  consists  of  gold,  sUv&t^  and  nickeL 

Gold*    The  20,  10,  and  5  mark  pieces. 

Silver.    The  2,  and  1  mark,  and  the  20-penny  pieces. 

Nickel.    The  10,  and  the  5-penny,  and  pieces  of  less  valuation. 

The  10-mark  piece  {gold)  is  equal  to  3J  P.  Thalers  (old). 

The  l-mark  {diver)  is  equal  to  10  S.  Groschen,  or  1000  pennies. 

The  20-penny  {silver)  is  equal  to  2  S.  Groschen,  or  ^  of  a  mark. 

The  10-penny  {nickel)  is  equal  to  1  S.  Groschen,  or  ^jj  of  a  mark. 

JAPAN    MONEY. 
93.  Japan  has  a  new  and  decimal  system  of  coinage. 
94.  The  Unit  of  Japan  money 
is  the  gold  Yen,  valued  at  $.997 
U.  S.  money. 

The  Coin  of  Japan  embraces  five  gold  coins,  valued  at  $20,  $10, 
$5,  $2,  and  $1.  Also  five  silver  coins,  valued  at  5,  10,  20,  50,  and 
100  cents,  respectively. 

The  weight  of  the  new  trade  dollar  is  420  gr.,  and  .9  pure  silver. 


MONEY. 


261 


96,  The  following  shows  the  manner  in 
of  foreign  exchange  are  made  in 
as  quoted  Jan,  2,  1875  : 


which  quotations 
this  country,  and 


Sixty 

4.85^4  ' 
4.85  ( 
4.84     ( 


London  Prime  Bankers'  Sterl,  Bills 

Do Good        do.  do. 

Do Prime  Commercial    do. 

Paris Francs 5.17^  © 

Antwerp Francs 5.17v^  @ 

Switzerland Francs. ..^ 5.17^  @ 

Amsterdam Guilders. 41^^  @, 

Hamburg* Reichsmarks 947^  @, 

Fi-ankfort Reichsmarks 94%  @ 

Bremen Reichsmarks 9i%  @ 

Berlin Reichsmarks 94%  @. 


4.86 

4.85X 

4.85 

5.I614 

5.I614 

5.I614 

41X 

951/8 

95>^ 

951/i 

95>^ 


TTiree  Days. 
4.90     ©  4.90>^ 
4.8914  ©  4.90 


4.88K2 
5.13X 
5.13^ 
5.139£ 

41X 

96 

96 

96 

96 


© 


4.893/ 

5.12y, 

5.12;^ 

5.121/2 

41% 

96X 

96X 

96X 

963^ 


In  the  above,  "  Prime  Bankers'  Bills"  are  those  on  the  most  reliable  banking 
houses;  "Good"  is  applied  to  those  of  somewhat  inferior  credit;  and  "Prime 
Commercial"  are  merchants'  drafts,  which  usually  command  a  less  price  in  the 
market.  The  quotations  in  the  Jirst  column  are  those  of  60-day  bills,  and  in 
the  second  column  those  of  3  days. 


97.  Rates  of  Exchange  at  London,  and  on  London. 


EXCHANGE  AT  LONDON,  JAN.  2, 1875. 

EXCHANGE  ON  LONDON. 

ON 

TIME. 

BATE. 

DATE. 

TIME. 

EATE. 

Amsterdam 

Antwero 

short. 

11.13/2@11-16)^ 

25.47>^@25  5214 
20.78    @20.82 
25.15    @25.25 
25.47)^@-25.52i/2 
11.37/2@11.42 
20.78    @20.82 
20.78    ©20.82 
32i/2@     32% 
483^(1^     48% 
52%@     525^ 
28.173^(??^28.22i/2 
28.17Xfr?^'28.22X 
28.17/2@'28.22^ 

Jan.  2. 

short. 

3  mo. 
short. 

11.82 
2517 

Hamburg 

20.25 

Paris 

short. 
3  months. 

25.19 

Paris 

Vienna 

Jan.  2. 

3  mo. 
short. 

11050 

Berlin 

6.24X 

Frankfort 

St.  Petersburg. . 

my, 

Cadiz 

Lisbon 

90  days. 
3  months. 

Mil  m .... 

Genoa 

Naples 

• 

New  Yorl?:. . 

Dec.  31. 
Dec.  17 

60  days. 
90  days. 

1^4  80 

Rio  de  Janeiro. 

26%  @  2614 

Buenos  Ayres.. 

Valparaiso 

Bombay 

Dec.  31 
Dec.  29. 
Dec.  24. 
Dec   25 

6  mo. 

Is.  iokd. 

Calcutta 

Is.  I0|;.d. 

Honir  Kong^ .  r. . 
Shanghai.. 

4s.    2i^d. 

5s.83i£d.  @,5s.9d. 

96% 

Alexandria. 

Dec.  30. 

3  mo. 

262 


MONEY. 


98.     Weight,  Fineness,  and  Value  of  Foreign  Gold  Coins, 
as  determined  hy  United  States  Mint  Assays, 


Country. 


Austria 

Do 

Do 

Belgium 

Brazil 

Centr'l  America 

Do.       do. 

Chili 

Colombia   and 
S.  A.  generally. 

Denmark 

Egypt 

England 

Do 

France 

Germany 

Greece  

India  (British). 

Italy 

Japan 

Do 

Mexico 

Do 

Do 

Netherlands  ... 
New  Granada.. 

Peru 

Portugal 

Russia 

Spain 

Do 

Do 

Sweden 

Do 

Tunis 

Turkey 


Denomination. 


Fourfold  ducat , 

Souverain  (no  longer  coined) 

4  florins 

25  francs 

20  milreis 

2e8cudos 

4  reals 

10  pesos  (dollars)  . .  

Old  doubloon 

Old  10  thaler 

Bedidlik  (100  piasters) 

Pound  or  Sovereign  (new) . . 

Pound  average  (worn) 

20  franc  (no  new  issues) 

Old  10  thaler  (Prussian)    . . . 

20  drachms 

Mohur,  or  15  rupees   

20  lire  (francs) 

Cobang  (obsolete) 

New  20  ven 

Old  doubloon  (average) 

20  pesos  (empire) 

20  pesos  (republic),  new 

10  guilders 

10  pesos  (dollars) 

20  soles 

Coroa  (crown) 

5  roubles 

100  reales 

80  reales 

10  escudos 

Ducat 

Carolin  (10  francs) 

25  piasters 

100  piasters 


Weight. 


0.448 

0.363  ♦ 

0.104 

0.254 

0.575 

0.209 

0.027 

0.492 

0.867 

0.427 

0.275 

0,256,8 

0,S50,3 

0,307 

0.427 

0.185 

0.375 

0.207 

0.289 

1.072 

0.867 

1.0P6 

1.081 

0.215 

0.525 

1.055 

0.308 

0.210 

0.268 

0.215 

0.270,8 

0.111 

0.104 

0.161 

0.231 


Fineness. 


TJwus'dths. 
986 
900 
9C0 
899 
916,5 
853,5 
875 


870 

895 

875 

916,5 

916,5 

899 

903 

900 

910,5 

899 

572 

900 

870 

875 

873 

809 

891,5 

898 

912 

916 

896 

869,5 

896 

975 

900 

900 

915 


Value  in  U.S. 
gold  coin. 


$  cts.m. 
9    13 

75 

93 

72 


4 
5 
0 
4 
8 

48.  8 
13    6 


15    50    3 


19    21 


80 


96 


7  10 

3  84 

3  57 

19  94 

15  59    3 

19  64    3 

19  51    5 

3  99    7 
67    5 


97    6 


5 
3 
4 
3 
5 
2 

1  93 

2  99 
4    37    0 


1.  Foreign  gold  coins,  if  converted  into  United  States  coins,  are 
subject  to  a  charge  of  one-fifth  of  one  per  cent. 

2.  For  sU^er  coins  there  is  no  fixed  legal  valuation,  as  compared 
with  gold.  The  value  of  the  silver  coins  January  1, 1874,  wa«  com- 
puted at  the  rate  of  120  cents  per  ounce,  900  fine,  payable  in  sub- 
sidiary  silver  coin,  or  113  cents  in  gold. 


TABLE     FOR     INVESTORS. 

99.  The  following  liable  shows  the  rate  per  cent,  of  Annual  Income 
from  Bonds  hearing  5,  6,  7,  or  8  per  cent,  interest,  and  costing 
from  40  to  125. 


Purchase 
Price. 

5%. 

6%. 

7/.. 

8%. 

Purchase 
Price. 

5f^. 
6.02 

6%. 

7%. 

8%. 

40 

12.50 

15.00 

17.50 

20.00 

83 

7.22 

8.43 

9.63 

41 

12.20 

14:64 

17.08 

19.52 

84 

5.95 

7.14 

8.33 

9.52 

42 

11.90 

14.28 

16.66 

19.04 

8.^ 

5.88 

7.05 

8.23 

9.41 

43 

11.63 

13.95 

16.28 

18  61 

86 

5.81 

6.97 

8.13 

9.30 

44 

11.36 

13.63 

15.90 

18.18 

87 

5.74 

6.89 

8.04 

9.19 

45 

11.11 

13.32 

15.56 

17.78 

88 

5.68 

6.81 

7.94 

9.09 

46 

10.86 

13.04 

1521 

17  39 

89 

5.61 

674 

7.86 

8.98 

47 

10.63 

12.77 

14.90 

17.02 

90 

5.55 

6.66 

7.77 

8.S8 

48 

10.41 

12.50 

1453 

16.66 

91 

5.49 

6.59 

7.69 

8.79 

49 

10.20 

12.25 

14.29 

16.33 

92 

5.43 

6.52 

7.60 

8.69 

50 

10.00 

12.00 

14.00 

i6.o:) 

93 

5.37 

6.45 

7.52 

8.60 

51 

9.80 

11.73 

13.73 

15.68 

94 

5.31 

6.38 

7.44 

8.51 

52 

9.61 

11.53 

13.46 

15.38 

i   95 

5.26 

6.31 

7.36 

8.42 

53 

9.43 

11.32 

13.20 

15.09 

!   96 

5.20 

6.25 

7.29 

8.33 

54 

9.25 

11.11 

12.96 

14.81 

;   97 

5.15 

6.18 

7.21 

8.24 

55 

9.03 

10.90 

12.72 

14.54 

!   98 

5.10 

6.12 

7.14 

8.16 

56 

8.92 

10?70 

12.50 

14.28 

1   99 

505 

6.06 

7.07 

8.08 

57 

8.77 

10.52 

12.27 

14.03 

1  100 

5.00 

6.00 

7.00 

8.00 

58 

8.62 

10.34 

12.06 

13.79 

101 

4.95 

5.94 

6.93 

792 

59 

8.47 

10.16 

11.86 

13.55 

102 

4.90 

5.88 

6.86 

7.84 

60 

8.33 

1000 

11.66 

1-3.33 

103 

485 

5.82 

6.79 

7.76 

61 

8.19 

9.83 

11.47 

13.11 

104 

4.80 

5.76 

6.72 

7.69 

62 

8.06 

9.67 

11.20 

12.90 

105 

4.76 

5.71 

6.66 

7.61 

63 

7.93 

9.52 

nil 

12.69 

106 

4.71 

5.66 

6.60 

7.54 

64 

7.81 

9.37 

10.93 

12.50 

107 

4.67 

5.60 

6.54 

7.47 

65 

7.69 

9.23 

10.76 

12.30 

108 

4.62 

5.55 

6.48 

7.40 

m 

7.57 

9.09 

10.60 

12,12 

109 

4.58 

5.50 

0.42 

7.33 

67 

7.46 

8.95 

10.44 

11.94 

no 

4.54 

5.45 

6.36 

7.27 

68 

7.35 

8.82 

10.29 

11.76 

111 

4.50 

5.40 

6.30 

7  20 

69 

7.24 

8.69 

10.14. 

11.53 

112 

4.46 

5.35 

6.25 

7.14 

70 

7.14 

8.57 

10.00 

11.43 

113 

4.42 

5.30 

6.19 

707 

71 

7.04 

8.45 

9.85 

11.26 

114 

4.38 

5.26 

6.14 

7.01 

72 

6.94 

8.33 

9.72 

11.11 

115 

4.35 

5.21 

6.08 

6.95 

73 

6.84 

8.21 

9.58 

10.95 

116 

4.31 

5.17 

6.03 

689 

74 

6.75 

8.10 

9.45 

10.80 

117 

4.27 

5.12 

5.98 

6  83 

75 

6.66 

8.00 

9.33 

10.66 

118 

4.23 

5.08 

5.93 

6.77 

76 

6.57 

7.89 

9.21 

1052 

119 

4.20 

5.04 

5.88 

6.';  2 

77 

6.49 

7.79 

9.00 

10.38 

120 

4.16 

5.00 

5.83 

6.66 

78 

6.41 

7.69 

8.97 

10.25 

121 

4.13 

4.95 

5.78 

6.61 

79 

6.32 

7.59 

8.8o 

10.12 

122 

4.09 

4.91 

5.73 

6.55 

80 

6  25, 

7.50 

8.75 

10.00 

123 

4.03 

4.87 

5.69 

6  50 

81 

6.17 

7.40 

8.64 

9.87 

124 

4.03 

4.83 

5.65 

6.45 

82 

6.09 

7.31 

8.53 

9.75 

125 

4.00  ' 

4.80 

5.60 

6.40 

.-A 

264 


31  0  ]sr  E  Y  . 


STATUTE    LIMITATIONS. 

100.  A  forced  collection  of  debts  cannot  be  made  after 
a  certain  number  of  years  specified  in  tlie  statute  of  limita- 
tions of  the  different  States  named  in  the  following  Table  : 


Name  of  States. 


Alabama 

Arkansas 

California.. 

Connecticut 

Colorado 

Delaware 

Dist.  of  Columbia. 

Florida 

Georgia 

Illinois 

Indiana 

Iowa 

Kentucky 

Kansas 

Louisiana 

Maine 

Maryland 

Massachusetts 

Michigan 


*i 

aj 

§3 

S 

a 

^ 

^ 

Ha 

Yrs. 

Yrs. 

Yrs. 

3 

6 

20 

3 

7 

10 

2 

4 

10 

6 

6 

17 

2 

4 

5 

3 

6 

20 

3 

3 

12 

5 

5 

3 

3 

12 

5 

6 

16 

6 

20 

20 

5 

10 

20 

2 

7 

14 

3 

5 

10 

3 

5 

10 

6 

6 

20 

8 

3 

12 

6 

6 

20 

6 

6 

20 

Name  of  States. 


Minnesota 

Mississippi 

Missouri 

New  Hampshire. 

New  Jersey 

New  York 

North  Carolina.. 

Ohio 

Oregon 

Pennsylvania. 

Rhode  Island 

South  Carolina.. 

Tennessee 

Texas , 

Utah 

VennoDt 

Virginia 

West  Virginia... 
Wisconsin 


^* 

«  B 

<U   0 

S 

0,0 

^ 

08 

1 

Yrs. 

Yrs. 

6 

6 

3 

6 

5 

10 

6 

6 

6 

16 

6 

G 

3 

3 

6 

15 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

2 

4 

6 

6 

5 

5 

5 

5 

10 

' 

Yrs. 
10 
20 
20 
20 
20 
20 
10 
20 
10 
20 
20 
20 
10 
10 


0 
10 
10 
10 


1.  The  above  data  are  liable  to  a  cbange  at  any  time  by  the  Leg- 
islatures of  the  States  respectively. 
3.  In  some  of  the  above  States  there  are  exceptions  and  conditions. 

LEGAL    TENDER. 

101.  All  gold  coins,  of  United  States  coinage,  are  legal 
tender  in  payment  of  all  amounts. 

All  silver  coins  are  legal  tender  in  sums  not  exceeding 
Jive  dollars,  in  any  one  payment. 

The  five-cent,  three-cent,  and  one-cent  coins  are  legal 
tender  at  their  nominal  value,  in  sums  not  exceeding 
twenty-five  cents,  in  any  one  payment. 

^^  GreenbacTcs  "  are  legal  tender  in  payment  of  all  debts 
public  and  private,  except  duties  on  imports,  and  interest 
on  the  public  debt. 


MONEY. 


265 


103.    COMPOUND     INTEREST     TABLE. 

Amount  at  the  end  of  the  year,  of  One  Dollar  per  annum  i^paid  in 
advance),  at  Compound  Interest /(?r  any  number  of  years. 


Yrs. 

3  per  cent. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

7  per  cent. 

8  per  cent. 

1 

$1.03 

$1.04 

$1.05 

$1.06 

$1.07 

$1.08 

2 

2.09 

2-12 

2.15 

2.18 

2  21 

2.25 

3 

3.18 

3.25 

3.31 

3.37 

3.44 

3.51 

4 

4.31 

4.42 

4  53 

4.64 

4.75 

4.87 

5 

547 

5.63 

5  80 

5.98 

6.15 

6.34 

6 

^m 

6.90 

7.14 

7.39 

7.65 

7.92 

7 

7.89 

8.21 

8.55 

8.90 

9.27 

9.64 

8 

9.16 

9.58 

10.03 

10.49 

10.98 

11.49 

.9 

10.46 

1101 

11.58 

12.18 

12.82 

13.49 

10 

11.81 

1249 

13.21 

13.97 

14.78 

15.65 

11 

13.19 

14.03 

14.92 

15.87 

16.89 

1798 

13 

1462 

15.63 

16.71 

17.88 

19.14 

20.50 

13 

16.09 

17.29 

18.60 

20.02 

21.55 

2321 

14 

17.60 

19.02 

20  58 

22.28 

24.13 

26.15 

15 

19.16 

20.82 

22.66 

24.67 

26.89 

29.32 

16 

2076 

22.70 

24.84 

27.21 

29.84 

82.75 

17 

22.41 

24.65 

27.13 

30.00 

33  00 

36.45 

18 

24.12 

26.67 

29.54 

32.76 

36  38 

40.45 

19 

25.87 

23.78 

32.07 

35.79 

40.00 

44.76 

20 

27.68 

30.97 

34.72 

38.99 

43  87 

49.42 

21 

29.51 

33.25 

3751 

42.39 

48.01 

54.46 

22 

31.45 

35.62 

40.43 

46.00 

52.44 

59.89 

23 

33.43 

38.08 

43.50 

49.82 

57.18 

65.76 

24 

35  46 

40.65 

46.73 

53.86 

62.25 

72.11 

25 

37.55 

43  31 

50.11 

58.16 

67.68 

78.95 

26 

39  71 

4^108 

5367 

62.71 

73.48 

86.35 

27 

41.93 

48.97 

57.40 

67.53 

79.70 

94.34 

28 

44.22 

51.97 

61.32 

72.64 

86.35 

102.97 

29 

46.58 

55.08 

65.44 

78.06 

93.46 

112.28 

30 

49.00 

58.33 

69.76 

83.80 

10107 

122.35 

31 

51.50 

61.70 

74.30 

89.89 

109.22 

133.21 

32 

54.08 

65.21 

79.06 

96.34 

117.93 

144  95 

33 

56.73 

68.86 

84.07 

103.18 

127.26 

157.63 

34 

59.46 

72.65 

89.32 

110.43 

137.24 

171.82 

35 

62  28 

76.60 

94.84 

118.12 

147.91 

186.10 

36 

65.17 

80.70 

100.63 

126.27 

159.34 

202.07 

37 

68.16 

84.87 

106.71 

134.90 

171.56 

219.32 

38 

71.23 

89.41 

113.10 

144.06 

184.64 

237.94 

39 

74.40 

94.03 

119.80 

153.76 

198.64 

258.06 

40 

77.66 

98.83 

126.84 

164.05 

213.61 

279.78 

41 

81.02 

103.82 

134.23 

174.95 

229.63 

303.24 

42 

84.48 

109.01 

141.99 

186.51 

246.78 

328  58 

43 

88.05 

114.41 

150.14 

198.76 

265.13 

355.95 

266 


MONEY, 


103. 


CARLISLE    TABLE. 

the  values  of  Annuities  on  Single  Lives,  according  to 
the  Carlisle  Table  of  Mortality. 


Agk.  4  per  cent.   5  per  ct. 


14.28164 
16.55455 
17.72616 
18.71508 
19.23133 

19.59203 
19.74502 
19.79019 
19.76i43 
19.69114 

19.58339 
19.45357 
19.33493 
19.20937 
19.08182 

18.95534 

18.83036 
18.72111 
18.G0G56 
18.48649 

18.36170 

18.23196 
18.093S6 
17.95010 
17.80058 

17.64486 

17.485S6 
17.32023 
17.15412 
16.99683 

16.85215 
16.70511 
16.55246 
16.39072 
16.21943 

16.04123 
15.85577 
15.66586 
15.47129 
15.27184 

15.07363 
14.83314 
14.69466 
14.50529 
14.30874 


12.083 
13.995 
14.983 
15.824 
16.271 

16.590 
16.735 
16.790 
16.786 
16.742 

16.609 
16.581 
10.494 
16.406 
16.316 

16.227 
10.114 

16.0G6 

15.987 
15.904 

15.817 
15.726 
15.628 
15.525 
15.417 

15.303 
15.187 
15.065 
14.942 

14.827 

14.723 
14.617 
14.506 
14.387 
14.260 

14.127 
13.987 
13.843 
13.695 
13.542 

13.390 
13.245 
13.101 
12.957 
12.806 


5  per  ct. 


10.439 
12.078 
12.925 
13.652 

14.042 

14.325 

14.400 
14.518 

14.526 
14.500 

14  448 
14.384 
14.321 
14.257 
14.191 

14.126 

14.007 
14.012 
13  956 
13.897 

13.805 
13.769 
13.697 
13.621 
13.541 

13.456 

13.308 
13.275 
13.182 
13.096 

13.020 
12.942 
12.860 
12.771 
12.675 

12.573 

12.465 
12.354 
12.239 
12.120 

12.002 
11.890 
11.779 
11.608 
11.551 


7  per  ct. 


9.177 
10.605 
11.342 
11.978 
12.322 

12.574 
12.098 
12.756 
12.770 
12.704 

12.717 

12.069 
12.621 
12.572 
12.522 

12.473 

12.^129 
12.309 
12.31-8 
12.305 

12.259 

12.210 
12.156 
12.098 
12  037 

11.972 

11.904 
11.832 
11.7o9 
11.693 

11.636 
11.578 
11.516 
11.448 
11.374 

11.295 
11.211 
11.124 
11.033 
10.939 

10.845 
10.757 
10.671 
10.585 
10.494 


Age. 

45 

4  per  ct. 

5  per  ct. 

14.10460 

12.648 

46 

13.88928 

12.480  I 

47 

13.66208 

12.301 

48 

13.41914 

12.107 

49 

13.15312 

11.892 

50 

12.86902 

11.660 

51 

12.56581 

11.410 

52 

12.25793 

11.154 

53 

11.94503 

10.892 

54 

11.62673 

10.624 

55 

11.29961 

10.347 

56 

10.96607 

10.063 

57 

10.62559 

9.771 

58 

10.28C47 

9.478 

59 

9.96331 

9.199 

60 

9.66333 

8.940 

61 

9.39309 

8.712 

62 

9.13376 

8.487 

63 

8.87150 

8.258 

64 

8.59330 

8.016 

65 

8.30719 

7.765 

66 

8.00900 

7.503 

07 

7.69080 

7.227 

68 

7.37970 

6.941 

69 

7.04881 

6.643 

70 

6.70936 

6.336 

71 

6.35773 

6.015 

72 

6.02548 

5.711 

73 

5.72465 

5.435 

74 

5.45P12 

5.190 

75 

5.2;3901 

4.989 

76 

5.02399 

4.792 

77 

4.82473 

4.609 

78 

4.62166 

4.422 

79 

4.39345 

4.210 

80 

4.18289 

4.015 

81 

3.95309 

3.799 

82 

3.74034 

3.606 

83 

3.53409 

3.406 

84 

3.32856 

3.211 

85 

3.11515 

3.009 

86 

2.92831 

2.8:30 

87 

2.77593 

2.685 

88 

2.68337 

2.597 

89 

2.57704 

2.495 

11.428 
11.296 
11.154 
10.998 
10.823 

10.631 
10.422 
10.208 
9.988 
9.761 

9.524 

9.280 
9.027 
8.772 
8.629 

8.804 

8.108 
7.913 
7.714 
7.502 

7.281 
7.049 
6.803 
6.540 
6.277 

5.998 
5.704 
5.424 
5.170 
4.944 

4.760 
5.579 
4.410 
4.238 
4.040 

3.858 
3.656 
3.474 
3.286 
3.102 

2.909 
2.739 
2.599 
2.515 
2.417 


MONEY. 


267 


ANNUITY    TABLE. 

104.  Showing  the  present  worth  of  an  Annuity  of  One  Dollar 
per  annumfiy  at  Compound  Interest,  from  1  year  to  Jfiy  inclusive. 


i 

3  per  ct. 

3.}  per  ct. 

4  per  ct. 

5  per  ct. 

6  per  ct. 

7  per  ct. 

1 

0.970  874 

0.966  184 

0.961  538 

0.952  381 

0.943  396 

0.934  579 

2 

1.913  470 

1.899  694 

1.886  095 

1.859  410 

l.a33  393 

1.808  017 

3 

2.828  611 

2.801  637 

2.775  091 

2.723  248 

2.673  012 

2.624  314 

4 

3.717  098 

3.673  079 

3.629  895 

3.545  951 

3.4C5  106 

3.387  209 

5 

4.579  707 

4.515  052 

4.451  822 

4.329  477 

4.212  364 

4.100  195 

6 

5.417  191 

5.328  553 

5.242  137 

5.075  692 

4.917  324 

4.766  537 

7 

6.230  283 

6.114  544 

6.002  055 

5.786  373 

5.582  381 

5.389  286 

8 

7.019  692 

6.873  956 

6.732  745 

6.463  233 

6.209  744 

5.971  295 

9 

7.783  109 

7.607  C87 

7.435  332 

7.107  822 

6.801  692 

6.515  228 

10 

8.530  203 

8.316  605 

8.110  896 

7.721  735 

7.360  087 

7.C23  577 

11 

9.252  624 

9.001  551 

8.760  477 

8.306  414 

7.886  875 

7.498  699 

12 

9.951  OW 

9.663  334 

9.305  074 

8. 863  2C2 

8.383  844 

7.942  671 

13 

10.634  955 

10.302  788 

9.905  648 

9.393  573 

8.852  683 

8.&57  685 

14 

11.296  073 

10.920  520 

10.563  123 

9.898  641 

9.294  984 

8.745  452 

15 

11.937  935 

11.517  411 

11.118  3G7 

10.379  658 

9.712  249 

9.107  898 

16 

12.561  102 

12.094  117 

11.652  206 

10.837  770 

10.105  895 

9.446  63^ 

17 

13.166  118 

12.051  321 

12.105  6C9 

11.274  0C6 

10.477  260 

9.763  20» 

18 

13.753  513 

13.109  602 

12.659  207 

11  689  587 

10.827  603 

10.059  070 

19 

14.323  799 

13.709  837 

13.133  9S9 

12.085  821 

11.158  116 

10.335  578 

20 

14.877  475 

14.212  403 

13.590  326 

12.462  210 

11.469  421 

10.593  997 

21 

15.415  024 

14.697  974 

14.029  IGO 

12.821  153 

11.764.077 

10.835  527 

22 

15.936  917 

15.167  125 

14.451  115 

13.163  003 

12.041  582 

H.C61  241 

23 

16.443  608 

15.620  410 

14.856  842 

13.488  574 

12.303  379 

11.272  187 

24 

16.935  542 

16.058  308 

15.246  963 

13.798  642 

12.550  358 

11.469.334 

25 

17.413  148 

16.481  415 

15.622  080 

14.093  945 

12.783  356 

11.653  5^3 

26 

17.876  842 

16. 890  352 

15.982  769 

14.275  185 

13,C03  166 

11.825  779 

27 

18.327  031 

17.285  365 

16.329  586 

14.64^  034 

13.210  534 

11.986  709 

28 

18.764  108 

17.667  019 

16.663  063 

14.898  127 

13.406  164 

12.137  111 

29 

19.188  455 

18.035  767 

16.983  715 

15.141  074 

13.590  721 

12.277  674 

30 

19.600  441 

18.392  045 

17.292  033 

15.372  451 

13.764  831 

12.409  041 

31 

20  000  428 

18.736  276 

17.588  494 

15.592  811 

13.929  086 

12.531  814 

32 

20.338  766 

19.068  865 

17.873  552 

15.802  677 

14.084  043 

12.646  555 

33 

20.765  792 

19  390  208 

18.147  646 

16.002  549 

14.230  280 

12.753  790 

ai 

21.131  837 

19.700  684 

18.411  198 

16.192  204 

14.368  141 

12.854  m 

35 

21.487  220 

20.000  661 

18.664  613 

16.374  194 

14.498  246 

12,947  67^ 

36 

21 .832  252 

20.290  494 

18.908  282 

16.546  852 

14.620  987 

13.035  208 

37 

22.167  235, 

70.570  525 

19.142  579 

16.711  287 

14.736  780 

13.117  017 

33 

22.492  462 

20.841  087 

19.367  864 

16.867  893 

14.846  019 

13.193  473 

39 

22.808  215 

21.102  500 

19.584  485 

17.017  041 

14.949  075 

13.264  928 

40 

23.114  772 

21.355  072 

19.792  774 

17.159  086 

15.046  297 

13.331  709 

2C8  MISCELLANEOUS. 

SPECIFIC    GRAVITIES.— WATER   1. 

105.  A  Table  showing  the  weight  of  each  substance  compared  with 
an  equal  volume  of  pure  water,  A  cubic  foot  of  rain-water  weighs 
1000  ounces,  or  62^  lb.  Avoir.  To  find  the  weight  of  a  cubic  foot 
of  any  substance  named  in  the  table,  remove  the  decimal  point  three 
places  toward  the  right,  which  is  multiplying  by  1000,  and  the  result 
icill  show  the  number  of  ounces  in  a  cubic  foot. 


Substances. 


Acid,  acetic  

"     nitric 

"     sulphuric 

Air 

Alcoliol,  of  commerce. . 

"        pure    

Alder  wood 

Ale 

Alum ,.., 

Aluminum 

Amber 

Amethyst 

Ammonia ,  

Ash 

Blood,  human , 

Brass (about) 

Brick 

Butter 

Cedar 

Cherry 

Cider 

Coal,  bituminous  (about) 

"     anthracite 

Copper 

Coral 

Cork 

Diamond 

Earth  (mean  of  the  globe) 

Elm 

Emerald 

Fir 

Glass,  flint 

"      plate 

Gold,  native 

"     pure,  cast 

*'     coin 

Granite 

Gnm  Arabic 

Cypsum 

]loney 

Ice 

Iodine 

Iron 

'•  ore 

Ivory 

Lard 


Specific  Grav 


1.008 

1.271 

1.841  to  2.125 

.001227 

.835 

.794 

.800 
1.035 
1.724 
2  560 
1.064 
2.750 

.875 

.800 
1.054 

.800 
2.000 

.942 
.457  to  '.561 

.715 
1.018 
1.250 
1.500 
8.788 
2.540 

.240 
8.530 
5.210 

.671 
2.678 

.550 

2.760 

2.760 

15.600tol9500 

19.258 

17.647 

2.652 

1.452 

2  288 

1.45B 

.930 
4.948 
7.645 
4.900 
1.917 

.947 


Substances. 


Lead,  cast 

"     white 

"     ore 

Lignum  vitae 

Lime 

"'    stone 

Mahogany 

Manganese 

Maple 

Marble 

Men  (living) 

Mercury,  pure 

Mica 

Milk 

Nickel 

Nitre 

Oil,  castor 

''    Unseed 

Opal  

Opium 

Pearl 

Pewter 

Platinum  (native) 

''  wire 

Poplar    

Porcelain 

Quartz 

Rosin 

Salt  

Sand 

Silver,  cast 

"      coin 

Slate  

Steel 

Stone 

Sulphur,  fused 

Tallow 

Tar    

Tin    

Turpentine,  spirits  of 

Vinegar  

Walnut 

Water,  distilled 

"       sea 

Wax 

Zinc,  cast 


Specific  GraVo 


11.350 
7.235 
7.250 
1.333 

.804 
2.386 
1.063 
3.700 

.750 
2.716 

.891 
14.000 
2.750 
1.032 
8.279 
1.900 

.970 

.940 
2.114 
l.:337 
2.510 
7.471 
17.000 
21.041 

.383 

2.385 

2.500 

1.100 

2  130 

1.500  to  1800 

10.474 

10.534 

2.110 

7.816 

2.000  to  2.700 

1.990 

.941 
1.015 
7.291 

.870 
1.013 

.671 
1000 
1.0C8 

.897 
7.190 


MISCELLANEOUS. 


269 


106.   ABBREVIATIONS  USED  IN  BUSINESS^ 


@ 

At. 

Guar. 

Guarantee. 

%  or  Acc't 

Account. 

Gal. 

Gallon. 

Am't 

Amount. 

Hhds. 

Hogsheads. 

Ass'd 

Assorted. 

Ins. 

Insurance. 

BaL 

Balance. 

Inst. 

This  month. 

Bbl. 

Barrel 

Invt. 

Inventory. 

Blk. 

Black. 

Int. 

Interest. 

B.  L. 

Bill  of  Lading. 

Mdse. 

Merchandise. 

f  or  ct 

Cents. 

Mo. 

Month. 

% 

Per  cent. 

Net. 

Without  disc't 

Co. 

Company. 

No. 

Number. 

Cr. 

Creditor. 

Pay't 

Payment. 

Com. 

Commission. 

Pd. 

Paid. 

Cons't 

Consignment. 

Pk'gs 

Packages. 

Dft. 

Draft. 

Per 

By. 

Disc't 

Discount. 

Prem. 

Premium. 

Do. 

The  same. 

Prox. 

Next  month. 

Doz. 

Dozen. 

Ps. 

Pieces. 

Dr. 

Debtor. 

Rec'd 

Received. 

Ea. 

Each. 

Ship't 

Shipment. 

Exch. 

Exchange. 

Sund's 

Sundries. 

Exps. 

Expenses. 

S.  S. 

Steamship. 

FeL 

Folio. 

Ult. 

Last  month. 

Fw'd 

Forward. 

Yd. 

Yards. 

Fr't 

Freight. 

Yr 

Year. 

4       5       7 
16  doz.  — --,  ^,  —  =  16  dbz.,  4  of  which  are  at  $10  per  doz.,  5 

%  $13,  and  7  @  $15. 
9  doz.,  f  (^  5/  f  @4/6.     3  doz.  No.  4  @  5  shillings  per  doz.,  and 

6  doz.  No.  5  @  4  shillings  sixpence  per 

doz. 
8  X  10,  or  8  by  10  in.    8  inches  wide  and  10  inches  long. 


270  MEASURES, 


FEENCH  AND   SPANISH   MEASURES. 

958.  The  old  French  Linear ^  and  Land  Meas- 
nre^  is  still  used  to  some  extent  in  Louisiana,,  and  in 
other  French  settlements  in  the  United  States. 

Table. 

12  Lines    =  1  Inch.  6  Feet      —  1  Toise. 

12  Inches  =  1  Foot.  32  Toises  =  1  Arpent. 

900  Square  Toises  =  1  Square  Arpent. 

The  French  Foot  equals  12.8  inches,  American,  nearly. 

The  Arpent  is  the  old  French  name  for  Acre^  and  contains  nearly 
f  of  an  English  acre. 

In  Texas,  New  Mexico,  and  in  other  Spanish  settle- 
ments of  the  United  States,  the  following  denominations 
are  still  used : 

Table. 

1000000  Square  Varas  =  1  Labor     =    177.136  Acres  (American). 
25  Labors  —  1  League  =  4428.4      Acres  " 

The  Spanish  Foot  =  11  11  +  in.  (Am.) ;  1  Vara  =  S3|  in.  (Am.); 
108  Varas  =  100  Yards,  and  1900.8  Varas  =  1  Mile. 

Other  Denominations  in  Use. 

5000        Varas  Square  =        1  Square  League. 
1000         Varas  Square  =        1  Labor,  or  ^V  League. 
5645.376  Square  Varas  =  4840  Square  Yards  =      1      Acre. 
23.76    Square  Varas  =        1  Square  Chain  =    ^    -^^  Acre. 
1900.8      Varas  Square  =        1  Section  =  640      Acres. 


In  many  answers  the  decimal  figures  following  the  second  or  third 
places  have  been  omitted,  and  when  the  first  figure  omitted  was  equal 
to,  or  greater  than  5,  the  last  figure  retained  was  increased  by  1 . 


Art.  512. 

2.  2431  lb. 


6d. 


9. 
10. 
11. 

12. 
13 


19, 


3.  $6321. 

4.  £263  2s. 

5.  2912  bu. 

6.  $175. 

7.  $205.49. 

8.  14.076  rd. 
$6014.40. 
$3180.01. 
2  mi.  277  rd. 

5V  ft. 

386|  ft. 

21  f  bu. 
U.  437i  lb. 
15.  123  men. 

17.  .004  hhd. 

18.  264|  lb. 

$9898.25. 
V.  $677,331  Ex. 

Savings, 
$922.66|. 
7.  .52  ;  $45760. 
'^.  $3903.40. 

Art.  515. 

2.  25%. 

3.  25%. 
108%. 
5%. 
14|%. 
5%. 
5|%. 

9.  62iy%.  ' 
to.  73A%. 

11.  m%. 

12.  7i%. 


4- 
5. 
6. 
7. 
8. 


75%. 

8%. 

37i%. 


13. 

u> 

15. 

16.  6%. 

17.  112i%. 

18.  20%. 

19.  121%. 

20.  50%. 

21.  65%. 

Art.  518. 


3.  $750. 

i.  91.2  A. 

5.  528  11). 

G.  690.    <9.  .6. 

7.  5800.  9.  100. 

i^*.  $750. 

11.  $5450. 

X?.  6r.0bu. 

i<f.  lOOSObbl. 

U.  8G00bu. 

15  4500  bu. 

16.  $3000. 

17.  $78133.33i. 

18.  $922.25. 

Art.  520. 

2.  2500. 

3.  $6000. 

5.  $1250. 

6.  S7400. 

7.  $3892.86. 

8.  36000. 

9.  $2275. 
i^.  900  bu. 
11.  800. 
i-^.  325  A. 


13.  $2480. 
i^.  $375.40. 
15.  $31  pr.  A. 
t?^  $4S  pr.  bale, 
i;2:--$4398.55. 

18.  $8750. 

19.  $3400,lstyr, 
$35^0,2dyr. 

20.  $208,331. 

Art.  530. 

2.  $349. 

3.  $842.40. 

4.  $636,375. 

5.  $204.86. 

6.  $253.75. 
$1437.60. 
$306.67. 
$144.32. 
$11016. 


8. 

9. 
10. 
11. 


Art.  531. 

3.  $208,125. 

4.  $11.31i. 

5.  $  17. 

6.  $6.22|-, 

7.  $4,375  ; 
$2.80. 

8.  $.53|-. 
$1.06i. 

9.  $.11|  per.lb 

Art.  532. 

3.  18f  %  gain. 

4.  12^%  loss. 

5.  20%. 

6.  28%. 


7. 

^. 

5. 
10. 
11. 

14|%, 

24%. 

66i%. 

23%. 

50%. 

12. 
13. 

37i%. 
60%. 

Art.  533. 

3. 
4. 
5. 
6. 
7. 
8. 

$9375. 

$8.80. 

$150. 

$14.14. 

$16666.66f. 

A.  $16000  ; 

B.  $10000. 

Art.  534. 

2. 

6.86.  3.  .75i 

4. 
5. 
6. 
7. 
<9. 
9, 

$4.91. 

$.20. 

$244,094. 

$183,331. 

$586.66f. 

$6553.60. 

Art.  535. 

2.  $1.47. 

3.  $150. 

4.  $1.03|. 

5.  $96. 

Art.  547. 

2.  $378,125. 

3.  $82.11. 

4.  $379.40. 

5.  $285.19. 


272 


ANSWERS. 


6.  $20.18. 

7.  $584.17^. 

8.  $96.90. 

Art.  548. 

S,  3i%.      S.  2f  %. 
S,  hfo.        6,  5%. 

Art.  549. 

^.  $2784.  5.  $9000. 
:^.  $3500.  6.   $960.40. 
4.  $9600. 

Art.  550. 

^.   $3750. 
^.  $583.33i 

4.  $25372. 

Art.  551. 

;^.  $4696.65. 

5.  $3182.55. 

4,  $1500. 

6.  $10648. 

^.  $6400.76  Inv. ; 
$320.04  Com. 

7.  31000  lb. 

8.  $10623.44. 

9.  $44231.71  Inv.; 
$1105.79  Com. 

10.   1640  yd. 

Art.  553. 

1.  48  bu. 
^.  $1700,  1st  yr.; 
$1785,  2d  yr. 

5.  24^\%. 

^.  $67 50  gain; 
12%  gain. 

5,  $3640. 

6,  $40842  cost. 
$6807  gain. 

7,  $30000. 
5.  40|%. 
5.  1468.75. 

10.  Loses  25%. 
/i.  25f  %  nearly. 
12.  5%. 
7.1  $4948.125. 
14.  $2964  whole  gain 
215  av.  gain  ^. 


i5.  Prints  @  $.15  ; 
Cassim.@$4.06i; 
Ticking  @  $.25 
Shawls  @  $9.20 
Thread  @  $.875  . 
Buttons®  $1.25; 
Amt.  (a)  $729.96. 

16.  $705.14. 

17.  $155.09. 

i<?.  61788.6  lb. + 

iP.  $.50. 

W.  $10582;  $132  Com. 

21.  5i%. 

22.  $8,875;  loss 4|%+. 
28.  $3049.20  whole 

gain; 
50%  gain  +. 

Art.  507. 

2.  $101.25  int. ; 
$551.25  amt.; 
$21  int.;  $471  amt. 

3.  $71.32  int.; 
$318.32  amt.; 
$16.47  int. ; 
$263.47  amt. 

4.  $208.33  int. ; 
$708.33  amt. ; 
$22.92  int. ; 
$522.92  amt. 

5.  $3.46  int.  at  6%. 
$4.03  int.  at  7%; 
$4.32  int.  at  7i%; 

6.  $115.70  at  5%; 
$185.12  int.  at8%; 
$208.26  int.  at  9%. 

7.  $196.41  int. at  6^%; 
$235.';0int.at7i%. 

8.  $58.97  int.  at  10%; 
$73.71  int.  at  12^% 

9.  $888.40  amt. 

10.  $71.87  amt. 

11.  $1176.50  amt. 
?.  $442.50 

Art.  509. 

?.  $12.58  int.  at  6% 
$8.89  at  4%. 

?.  $92.53(^5%; 
$148.04  @  8%. 


^. 


$269.47  @  7%; 
$288.72  @  74. 
$61.12  int. 
$292.50  int. 
$1204.12  amt. 
$276.52  amt. 
$41.27  Int. 
$421.99  amt. 
$85.72  Int. 
$13227.50. 

Art.  573. 

$22.70  Int. 

$3.84. 

$38.34. 

$242.94. 

$318. 

$269.34. 

Art.  574. 

$120.  5.  $82.36 
$.04.  6.  $10.96. 
$10.58. 

Art.  575. 

$58.93..4.  $159,745. 

$8.40.  5.  $67.09. 

$67.67.  6.  $38.11 

$8.63. 

$3647.61. 

$115.20. 

$1066.36. 

$2010.42. 

$142.45  +  . 

$1886.17. 

$131.40. 


$263.83. 

$828.07. 

$1936.60. 

$3925.17. 

$1120.69. 

$76.67. 

$1931.40  lossc 

Art.  577. 

$660,  $792. 

$6936.09. 

$6069.08. 


ANSWERS. 


273 


4.  $5l6;'ri. 

5.  $669.12 ;  $334.56. 

6.  $10000. 

Art.  579. 


$1500. 

$889.25. 
$650.80. 

Art.  581. 

7%. 
7%. 


to. 

11. 


4. 

5. 

6. 

7. 

8. 
10 
IL 
12. 
IS, 


2  %  a  month, 

lOAf.. 

25%;  16|%;   . 
12i%;  10%. 
100%;  40%; 
28i%;16|%;10% 

The   2d  is  J^%.^ 
better. 

Art.  nS'S. 

7  mo.  10  d. 

6  yr.  8  mo. 

7  mo.  6  da. 

3  yr.  4  mo.  24  da. 
33i  ;  20  ;  16| ; 
13| ;  10  yr. 

50;  40;  28f ; 
25  ;  16  yr. 
12i;  6i;  25  yr. 

Art.  586. 

$428.76. 

$189.15. 

$1 176. 14. 

$100.32. 

$4199  +  . 

$1495.77. 

$53.38. 

$1525.64. 

$1540  79. 

$987.23^' 

$1934.84. 

$18142.81. 


10. 


11. 


Art.  589. 

$464.10. 

$7308. 

$11.30. 

$1161.04.. 

$1047.52. 

Art.  597. 

$659.94. 

$30.14. 

$162.25. 

Art.  598. 

$312.50. 
$355.16. 

Art.  603. 

$281.83. 

$102.90. 

$1137.61. 

$43.65  in  favor  of 

dis. 
$931.20. 
$838.26. 

45tVo%. 

$931.83. 

$.05  per  bbl.  more 
profitable  to  buy 
at  $8.75  on  6  mo. 

$3677.75. 

Art.  615. 

$6.27  Bk.  dis. 
$591.23  proceeds. 
$1614.88. 
$10839  83. 
Mat.  Oct.  30 ; 
81   days  term  of 

dis.  ; 

$940.38  proceeds. 
Mat  April  8 ; 
46   days  term  of 

dis.  ; 

$t±!r  proceeds,  ff^, 
Mat.  Aug.  2 ; 
79   days  term  of 

dis. ; 
$1295.82  proceeds. 


8.  Mat.  Dec.  15  ; 
30  da.  term  of  dis. 
$1281. 77  proceeds. 

Art.  617. 

S.  $1434.20. 
3.  $719.61. 

$1951.03. 

$2291.44. 

$321.46. 

$659.88. 

$368.25 

Art.  619. 

$188.43  bal.  July 

1st. 
$4.90. 
$369.36. 

$327,927. 

Art.  648. 

$34256.25. 
$16856.25. 

$15843.75. 

Art.  649. 

250  shares. 
220  '' 
220   '' 

480   ** 
200  '' 

Art.  650. 

$25500.  4.   $693a 
$21100. 

Art.  651. 

8f%. 

8%  bonds  at  110 
If  %  better. 
6%  bonds  at  84. 
if  %  better. 

3tt%. 
Art.  652. 

62J.  4-  71f. 

33J%. 

75 ;  66f . 


5. 
6. 

7. 

2, 
3. 
6. 


5.  $40. 


274 


A  N  S  W  E  K  S  . 


Art.  053. 

2.  $5463.28. 

3.  $268.20. 

J^.  $262.66  better  to 
pay  in  currency. 

Art.  654:. 

2.  $4000;  $4035.87; 
$4109.59. 

3,  $74000. 
Jf.   $1755890. 

5.   Dim.  $28.25. 

G.  $113  per  annum. 

7.  Stock    invest,   is 

$50    better,    or 
ff%  yearly. 

8.  $21384  in  N.Y.S. 

6's; 
$42768  U.  S.  5's 
of  81. 

9.  $792. 

Art.  664. 

2,  $42.75. 

3,  $24.06. 
k.  $187.50. 
5.  $156.25. 

Art.  665. 

2,  1}%. 

^.  t%. 

Art.  666. 

2,  $13600. 


5.  $22220.77. 

(;.  $49147.91. 

7.  $24500. 

<S\  $24766.58.    ' 

d,  $9.90. 

Art.  675. 

2.  $284.78. 
S.  $1055.30. 


5.  $527.65. 

6.  $5888.57. 

7.  $4416.57. 

8.  $3263.93. 

9.  $1131.12  loss. 
10,  $7200. 

Art.  685. 

2.  $11350. 

3.  $19072.16. 
J^.  $401920. 

7.  $25.09. 

8.  $87.38. 
5.  $112.50. 

10.  $226.50. 

11.  .0228  tax  rate. 
$214.65. 

12.  $410.95. 

13.  $224.37. 
U.  $178.13. 
15.  $420900. 

Art.  700. 

2,  $1566.15. 

Jf.  ^-4764.84. 

5.  $5153.24. 

6.  $6388  80. 

7.  $5632.20. 

Art.  701. 

2.  $787.46. 

3,  $720. 

k,  $316.45. 

5.  451  sliares. 

<?.  97J%.     7  7  V/c 

7.  $20108.35.. 

Art.  706. 

.?.  $2003.25. 
3.  $3317.63. 
5.  $134.78. 
G.  $352.67. 

8.  $421.09. 

9.  $566.50. 

11.  $801.94. 

12.  $4621  16. 

13.  $5243.89. 
U.  $3500.40. 


Art.  707. 

2.  £1543  4s.  2(1. 

4.  2318.84  marks. 

5.  1664.13  marks. 

7.  31888.83  francs. 

8,  12918.75  francs. 

Art.  711. 

2.  $179.21. 

5.  5.31  francs. 

6.  $4,987. 

7.  £1055  12s.  4d.; 
£21  9s.  9.7d. 

8.  |e32.78  ind.  ex. 

9.  696.6  guild,  loss. 
10.  $12617.08. 

Art.  726. 

2.  $437.50. 

3.  $1703  25. 

4.  $1843.75. 

5.  $1234.88. 
G.  $63.18. 

7.  $5775. 
<?.  $.2376.28  duty. 
$6815.75   cost   in 
currency. 
9,  $1755.89. 
10.  $987.08. 

Art.  733. 

2.  3  mo.  25  da. 

S,  6  mo.  26  da.  time 

of  Cr.  ; 
June  27, 77  Eq.  time 
Jf..  May  5, 1875. 
5.  5  yr.  20  da.  from 
date  of  last  paym't 

7,  Nov.  26,  Eq.  time. 

8,  73  da.  term  of  Cr.; 
Feb.  26,  Eq.  time 

9,  Mar  7,  Eq.  time. 
$1178.01     cash 

value. 

Art.  734. 

2,  Aug.     19,    1875. 

Eq.  time. 

3.  June  7.  1876. 


A  I^  S  W  E  R  S  . 


275 


4.  June  27,  1874  ; 

Dis.  $149.28. 
6.  Apr.  23,  1874. 

6.  $233711^. 

7.  May  20,  1875. 

Art.  737. 

f .  Dec.  13,  Eq.  time. 

3.  Dec.  19. 

4.  Jan.  24,  1879. 

'Art.  738. 

5.  May  18 ; 
$1486.17  due. 

8.  Dec.  5,  1875. 

4.  $2069.59. 

5.  Oct.  27  ; 
$:il02  58. 

e.  $1272.33. 

Art.  739. 

S.  $2331.65  Sales  ; 
$762.83  Charges ; 
$1568.82  Net  pro- 
ceeds ; 
Bal.  due,  Dec.  27. 
S.  $3966.25  Sales ; 
$412.98  Charges ; 
$3553.27  Net  pro- 
ceeds ; 
Eq.  time  Apr.  14, 
1875. 

Art.  767. 

S.   60  bu. 

3.  $100. 

4.  $4.05. 

5.  44|  bbl. 

Art.  770. 

S.   9  horses. 

4.  100  yd. 

5.  16  men. 

6.  96  sheep. 

7.  $5355. 

8.  7  hr.  13i  min. 

9.  355  bu. 


10.  112imi. 

11.  od^  da. 
It   $7320. 

13.  9  yd. 

14.  $1M. 

15.  46  A.  134  P. 

16.  $63. 

17.  $10958.90. 

18.  $3.25. 

19.  $89.60. 
^0.   $120.  • 

21.  2  yr.  6  mo. 


Art.  772. 
43i-  tons. 
5 1  weeks. 
432  mi. 
15  da. 


Art.  774. 


^.  $498.08. 

4.  1120  bu. 

5.  $6428.57. 

6.  114^\  ream. 

7.  2201  Cd. 

8.  $52.79. 

9.  9  men. 
10.  546  bbl. 
ii.  2080  lb. 
12.  $100. 

2«?.  266605f  brick. 
14^  $236.25. 
i5.  694|  yd. 
16.  $1728. 
i7.  5  da. 

18.  150  yd. 

19.  3  yr.  4  mo.  24  da. 

20.  $11.66|. 
21:  9  men. 

22.  8.116  ft. 

23.  $48. 
^4.  $53.08. 
^5.  1.6  mo.  4- 

Art.  782. 

3.  A's  share  $320. 
B's      ''      $316. 

Cs     '^     $184. 


>^.  A.  $303.45. 

B.  $337.17. 

C.  $404.61. 

D.  $682.77. 

5.  A.  $1710. 
B.  $870.20. 

6.  A.  $6000. 

B.  $8402.25. 

C.  $:05575. 

D.  $3042. 

7.  $5785  20,  the  first; 
$5142.40,     the 

second. 

8.  $3516.80  A's  gain; 
$5861.33^  B's  '' 
$8205.861  C's  '' 

9.  $269559.55    Re- 

sources ; 

$26434.55     Lia- 
bilities ; 

$243125  Stock ; 

$125000    Origi- 
nal capital ; 

$118125  net  gain; 

$56700  Ames' 
share  * 

$37800  Lyon's 

Q  |"|  Ck  It*  fit     • 

$23625  Clark's 
share. 


Art.  783. 

2.  $2400  Barr  ; 
$2666.661  Banks ; 
$2933.33J  Butts. 

3.  $388,704+  A.; 
$249,169+  B.; 
$112,122  C. 

4.  $1344.164  A.; 
$2027.836  B. 

5.  $5700  A.; 
$3760  B. ; 
$1340  C. 

6.  $1688.434 

Crane ; 
$3868  862 

Childs  ; 
$2012.708  Coe. 


276 


ANSWERS. 


Art.  787. 

^.  $.32. 

5.  $.30  per  bushel. 

4.  $6  gain. 

6,  $6.16. 

Art.  788. 

5.  2  lb.  of  first  ; 

2  lb.  of  second ; 

3  lb.  of  third. 

4.  1  at  $4  ; 
5  at  $5 
3  at  $6 

1  at  $8, 

5.  3bb 
3  bbl.  at  $(}"; 

2  bbl.  at  $7|. 

6.  3  gal.  at  $1.20  ; 

3  gal.  at  $1.83  ; 

15  gal.  at  $2.30  ; 
8  gal.  water. 

Art.  789. 

e.  10  cows  at  $32  ; 

10  cows  at  $30  ; 

60  cows  at  $48. 
S.  10  lb.  at  $.80  ; 

10  lb.  at  $1.20  ; 

70  lb.  at  $1.80. 

4.  12  yd.  at  $3^  ; 

16  yd.  at  $li. 

5.  150  acres. 

Art.  790. 

j^.  30  men,  5  w^omen, 

20  boys. 
S.  33  g^  gal.  water. 

4.  16,  24,  4,  and  12 

da.  respectively. 

Art.  792. 

1.  72  and  48. 
^.  D^s  age  16  ; 

E's  age  24 ; 

F's  age  84. 

5.  15  bu. 
4,  18  da. 

6.  8|  da. 


Starch  $2  a  box  ; 
Soap  $3. 
8Ha.; 

First  in  26|  da.  ; 
Second  in  40  da. ; 
Third  in  20  da. ; 
$180  share  of  1st ; 
$120  share  of  2d  ; 
$240  share  of  3d. 
14  bbl.  at  $10  ; 
6  bbl.  at  $7. 
16  min.  21f'Y  sec. 

past  3  o'clock. 
Wheat  $1.33  J  per 

bu.; 
Oats  $.50  per.  bu. 
8  da. 
$347.71. 
50  bu. 

27%  nearly. 
$7384j^3  younger ; 
$11076i-f  elder. 
1461  ft. 


$960  first ; 
$  ?20  second ; 
$840  third. 
$1570.31. 
506  lb. 
Oct.  26, 1875. 


^7. 


$33345; 
$27359.999; 

$25106.82. 
$1.60. 
42  geese ; 
58  turkeys. 
$5700. 
$282.24  Sim.  Int. ; 
$2202.24  Amt. ; 
$295.56  Com.  Int. ; 
$2215.56  ''  Amt.; 
$1673.93-}-  Pres- 
ent Worth ; 
$246.07  True  Dis. ; 
$283.20  Bk.  Dis. ; 
$1636.80  Proc'ds; 
$2252.199  Face. 


28.  $315.79. 

$473  69. 
$710.52. 

29.  $900,  July  28. 
SO.   $.97|. 

31.  $10665.80  in  U.  S. 

6's,  5-20. 
$21331.60  in  U.S. 
5's  of  '81. 

32.  A.  3600  bu.; 

B.  1200  bu. ; 

C.  1200  bu.   • 

33.  $1.72. 

34.  fn- 

35.  $5614.27  Net 

Proceeds. 
July  10,  Eq.  time. 

36.  $6100  M.'s  Cap.; 
15  mo.  N.'s  time. 

37.  $2023.22 ;  Apr.  24. 

38.  $2244.66. 

Art.  802. 

2.  1369;  1764; 
3136;  5625. 

3.  3375  ;  5832  ; 
74088  ;  157464. 

4.  3969  ;  110592  ; 
1048576  ;  248832. 

/?   49  .   1728 

^-  ¥5  6"  >  "sgy^T* 

7  _a5iLl_  •  318S 
'  •  384  16  >   3  2'  • 

8.  645.16. 

9.  1191016. 
10.   1958jV. 

/  7       1  4  G  4  1 
■^■^'     50625- 

12.  .00116964. 

13.  .015625. 

14.  46733.803208. 

15.  .065528814274496. 

16.  33169  If. 

17.  16.6056I-. 

18.  24.76099. 

19.  .000000250047. 

20.  1520875. 

21.  2023//^. 

22.  5.887. 

23.  640000. 

24.  2540.0390625. 

25.  125.       26.  1200 


ANSWERS. 


277 


Art.  803. 

3.  1764. 

4.  2304. 

5.  3136. 
6'.  9604. 
7.  15625. 
<^.  11025. 
9,  50625. 

i^.  38809. 
lU  116964. 

Art.  804. 

;^.  39304. 

4.  110592. 

5.  262144. 

6.  857375. 

7.  1953125. 

Art.  810. 

^.  8  ;  16  ;  24  ; 

81. 
^.  9  ;  14  ;  21 ; 

15. 

Art.  813. 

S.  85. 

4.  242. 

5.  98. 
^.  115. 
7.109. 
8,  997. 
P.  1432. 

10,  5464. 
i^.  «. 
i5.  If 

15,  .035. 

m  14.0048  +  . 

17.  1.5005 +  . 

18.  7.625. 

19.  4.213 +  . 
^^  103.9. 
;^i.  59049. 

^^.  3.00001 654-. 
^S.  5.656854  +  . 
^^.  1.5411. 
^5.  .91287+; 

ee.  .04419. 


,V.  36.37. 
1"^.  1.50748 +  . 
^9.  64. 

^^.  tV- 
^i.  1. 

S2.  1.78 +  . 
^«f.  72. 
S4,  90. 
^5.  480.8827. 

Art.  815. 

i.  1008  ft. 
^.  240.33  rd. 
S,  52  rd. 

4.  200.56  rd. 

5.  145ird. 

6.  $187.20. 

Art.  819. 

«^.  25. 

4.  55. 

5.  101. 

6.  165. 

7.  1015. 
^.  1598. 

10.  a. 

11'  f  *• 
12.  1.42  +  . 
i<^.  34. 
i.^.  .45. 
15.  2.34. 
i^.  4624. 
17.  .0809. 
i^.  .7936. 
i5>.  5.73  +  . 

^i.  .5569. 

22,  1 

^<?.'  14.75. 

;^.4.  60  8. 

Art.  821. 

1.  3  ft. 

2.  8  ft. 
^.  2  ft. 

4.  12150  sq.ft. 

5.  5  ft.  8+  in. 

6.  Oft.  5.3  + in 

7.  8  ft.  1.4  in. 


Art.  822. 

2.  274. 

3.  32. 
4-  543. 

5.  1.05 +  . 


Art.  829. 

3.  8.  ^.  149, 

4.  17.  7.  16. 

5.  33.  8.  7J|. 

Art.  830. 

2.  2.        .5.  4. 
^.  2.        G.  7%, 
4^  -|.       7.  A. 

Art.  831. 

i*.  9. 

3,  15. 
4.4. 

5.  27. 

6.  11  yr. 


Art.  832. 

2. 
3. 

4. 
5. 
6, 

600. 
154. 
125000. 

78. 
57900  ft. 

Art.  840. 

4. 
5, 
6. 
7, 
8. 

6144. 
3. 

$524288. 
$315,619  + 
$10485.76. 

Art.  841. 

2.  i.       4.  5. 

3,  5.        5,  3. 

Art.  842. 

2.  9. 

3.  7. 

4.  8. 


Art.  843. 

3,  765. 

5.  16. 

7.  2. 

<?.  280. 

^.  $1023. 

10.  $5314.40. 

Art.  853. 

3.  $3819.75. 

4.  $1292.31. 

5.  $3625. 

6.  6  yr. 

7.  7%. 

8.  $375.30. 

Art.  854. 

«^.  $300. 

4.  $3907.665  + 

5.  $1182.05  +  . 

6.  $3725.87  +  . 

7.  $629,426  +  . 

Art.  882. 

2.  600  sq.  ft. 

3.  4:2j\  sq.  ft. 

4.  22  A.  6  sq. 

ch.  13.45  P. 

5.  $449.07. 

6.  $147. 

7.  210  sq.  ft. 

Art.  883. 

2.  4i  ft. 

3.  13  in. 

4.  28  rd. 

5.  672  rd. 

yd. 

6.  8i  ch. 

7.  50rd. 

Art.  884. 

2.  111.80  sq.  ft. 

3.  3  sq.  ft.  1.7 

sq.  in. 

4.  13  A.   41.76 

P. 

5.  349.07  sq.  ft 


5^ 


278 


AJq^SWERS. 


Art.  886. 

2.  39  ft. 

8.  25  ft.  7.34  in. 

J^,  33.97  ch. 

5.  28  ft.  3.36  in. 

Art.  887. 

2.  45  yd. 

3.  19  ft.  2.5  in. 
Jf.  360  ft.  6f  in. 

6.  20  ft.  . 

Art.  898. 

2.  84  sq.  ft. 

3,  5  J  A. 

Art.  899. 

2,  11178  sq.  ft. 

3,  28|  sq.  ft. 

4,  2  A. 

Art.  900. 

2,  213  sq.  ft. 

3,  17  A.  8  ch.  3.4  P. 

Art.  904. 

8,  15  ft.  10.98  in. 
.4.  5  ft.  10.67  in. 

5,  5  ft. 

6,  7  ft.  3.96  in. 

Art.  905. 

4.  318.3  A.  + 

5.  114.59  A. 

Art.  906. 

3.  7  rd. 

J^.  19.098  ft.  Diam. 
59.998  ft.  Circum, 

Art.  907. 

2.  141.42  ft. 
8.  23.4  yd.  + 
Jt-.  7.07  ft.  + 

Art.  908. 

2.  32.98  sq.  ft.  4- 

3.  796.39  sq.  ft. 

Jt.  1  A.  75.62  P.  land. 
78.54  P.  water. 


Art.  909. 

f .  84. 
3'  28. 

^.  A- 

5.  32  lb.  13.7  oz. 

Art.  910. 

5.  369  rd.  L.; 
123  rd.  W. 

6.  3.5  in. 

7.  221 ;    238 ;    and 

255  ft. 

8.  $75. 

9.  126.78  rd. 

Art.  911. 

1.  $185.53. 
^.  35.35  ft.  + 
3,  403.7  rd.  + 
.4.  $5812.50. 
5.  $32.40. 

^.  28.66  P.  + 

7.  5  A. ;  or  twice  as 

large. 

8.  $724.75.    - 

9.  20  ft. 

iry.  98  A.  28  P. 

11.  14.645  ft. 

12.  294 rd.;  45.36 rd. 

13.  14  A.  150.4  P. 
U.  6  in. 

Art.  918. 

.4.  207.34  sq.  ft. 
5-  168|  sq.  ft. 
e.  263.89  sq.  ft. 
7.  301.177  sq.  ft. 

Art.  919. 

3.  274|  cu.  ft. 
Jf.  $27. 

5.  73.63  cu.  ft. 
^.  $53.70. 

Art.  925. 

2.  824  67  sq,  ft. 

3.  429|  sq.  ft. 
.4.  512.9  sq.  ft. 

5.  $25. 


Art.  926. 

3.  39.27  cu.  ft. 

4.  $29.23. 

5.  192000  cu.  ft.  vol 
22284.6  sq.   ft. 

surface. 

Art.  927. 

2.  345  sq.  ft. 

3.  256|  sq.  yd. 

Art.  928. 

2.  58.1196  cu.  ft. 

3,  38i  cu.  ft. 
It,  64.99  cu.  ft. 

Art.  932. 

2.  28.27  sq.  ft. 

3,  12.57  sq.  ft. 

Art.  933. 

2.  14137.2  cu.  ft. 

3,  523.6  cu.  yd. 

Art.  934. 

2.  10  ft.;  15  ft.;  and 

20  ft. 

3.  24  ft.;  33  ft.;  and 

40  ft. 

Art.  936. 

i.  13.228  ft.  edge. 
2315.03  cu.  ft.  vol. 

2.  11  ft.  7  in. 

3.  1494.257  gal. 
It.  $5.46. 

5.   576  ft. 
^.  14.42  in. 
7.  40  sq.  ft.  7f '. 
^.  1  cu.ft.  vol.of  cube 
Icu.ft.  659.5  cu.  in. 
vol.  of  sphere. 
9.   9  lb. 

10.  5  hr.  26.4  min. 

11.  12  ft.  6.79  in. 

12.  53.855  bu. 

Art.  937. 

2.  99.144  gal. 

3.  120.09  gal. 


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